feat: Start implementing Rostamian's pages (#39)
Began with incenter.html, the first one alphabetically. Needed one new point construction method, and a new option to see what was going on. Got the planar diagrams on that page working. The next step on #36 will be to get 3D diagrams as the theorem on this page generalizes to 3D. That will be a bigger task, so merging this now. Reviewed-on: #39 Co-authored-by: Glen Whitney <glen@studioinfinity.org> Co-committed-by: Glen Whitney <glen@studioinfinity.org>
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@ -7,12 +7,17 @@
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<body>
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<h3>Debugging</h3>
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Trace the following to the JavaScript console:
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<h4>Java Geometry Applets</h4>
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Trace the following to the JavaScript console: <br/>
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<label for="commands">Commands executed</label>
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<input type="checkbox" id="commands">
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<label for="color">Colors assigned</label>
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<input type="checkbox" id="color">
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<br/>
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<br/>
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Alter execution of the translated applet: <br/>
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<label for="showall">Show all entities, even hidden ones</label>
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<input type="checkbox" id="showall">
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<script src="options.js" type="module"></script>
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</body>
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@ -11,6 +11,7 @@
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<li> <a href="inscribed-revived.html">Revived</a> </li>
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<li> <a href="inscribed-modified.html">After</a> </li>
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<li> <a href="jelts/book1/joyceDefI2.html">Book I Def 2</a> </li>
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<li> <a href="rostamian/">Dr. Rostamian's unconverted pages</a> </li>
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</ul>
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<h2>WRL Files</h2>
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<ul><li> <a href="wrl_2.html">Target two</a></li>
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@ -0,0 +1,393 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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<html>
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<head>
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<!-- fix buggy IE8, especially for mathjax -->
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<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>The incenter via algebra</title>
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<link rel="stylesheet" type="text/css" media="screen" href="style.css">
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<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
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MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
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</script>
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</head>
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<body style="visibility:hidden">
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<h1>A triangle's incenter via algebra</h1>
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<h2>… and extension to tetrahedra</h2>
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<table class="centered">
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<tr><td align="center">
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<applet code="Geometry" archive="Geometry.zip" width="400" height="300">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="The incenter via algebra">
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||||
<param name="e[1]" value="A;point;fixed;100,250">
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<param name="e[2]" value="B;point;fixed;300,250">
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<param name="e[3]" value="C;point;free;50,50;red;red">
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||||
<param name="e[4]" value="ABC;polygon;triangle;A,B,C">
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||||
<param name="e[5]" value="pt1;point;extend;A,C,C,B;none;none">
|
||||
<param name="e[6]" value="C';point;proportion;A,pt1,A,C,A,B,A,B;none;none">
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||||
<param name="e[7]" value="pt2;point;extend;B,A,A,C;none;none">
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<param name="e[8]" value="A';point;proportion;B,pt2,B,A,B,C,B,C;none;none">
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<param name="e[9]" value="pt3;point;extend;C,B,B,A;none;none">
|
||||
<param name="e[10]" value="B';point;proportion;C,pt3,C,B,C,A,C,A;none;none">
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||||
<param name="e[11]" value="AA';line;connect;A,A';none;none;magenta">
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<param name="e[12]" value="BB';line;connect;B,B';none;none;magenta">
|
||||
<param name="e[13]" value="CC';line;connect;C,C';none;none;magenta">
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<param name="e[14]" value="O;point;intersection;AA',BB'">
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<param name="e[15]" value="H;point;foot;O,A,B;none;none">
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<param name="e[16]" value="IC;circle;radius;O,H;none;none;black;none">
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</applet>
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</td></tr>
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<tr><td>
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<b>
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Drag $C$ to change the geometry.<br>
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Press “r” to reset the diagram to its initial state.<br>
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The incenter lies at the intersection of angle bisectors.
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</b>
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</td></tr></table>
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<h2>Statement of the problem</h2>
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<p>
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It is a fact of elementary geometry that the center of
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a triangle's incircle—known as its <em>incenter</em>—lies
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at the point of intersection of the
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triangle's angle bisectors. This leads to an elegant expression for
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the location of the incenter as a linear combination of the
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triangle's vertices, as stated in:
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<p>
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<b>Proposition 1:</b>
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<i>
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Let $a$, $b$, $c$ be the lengths of the
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sides opposite to the vertices $A$, $B$, $C$ of the triangle $ABC$.
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Then the incenter, $O$, is expressed as a linear combination of
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the vertices as:
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\[
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O = \frac{a}{p}A + \frac{b}{p}B + \frac{c}{p}C,
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\]
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where $p = a + b + c$ is the triangle's perimeter.
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</i>
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<p>
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To make sense of this statement, you need to know something about
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the algebra of points. The following capsule summary is all that's needed:
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<blockquote>
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<p>
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<b>The algebra of points:</b>
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Consider the points $A$ and $B$ and a variable $t$ that takes
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values in the range 0 to 1.
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If $T$ is a point on the segment $AB$
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such that $AT/AB = t$, we write $T = (1-t)A + tB$. Note that when
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$t=0$ we get $T=A$, and when $t=1$ we get $T=B$.
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As the value of $t$ ranges from 0 to 1, the point $T$ slides
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from $A$ to $B$.
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</blockquote>
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<h2>A preliminary proposition</h2>
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<p>
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The following elementary proposition is needed for the proof of
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Proposition 1. This one is a standard result and is
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likely to be found in Euclid's <em>Elements</em> but I haven't checked.
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<p>
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<i>PS:</i>
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After this web page was written,
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I found out that
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Proposition 2 appears in Euclid's <em>Elements</em> as
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<a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI3.html">Book VI, Proposition 3</a>
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with a simpler and more elegant proof!
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Nevertheless, I am retaining my original proof here as
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a not-so-elegant alternative.
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<p>
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<b>Proposition 2:</b>
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<i>
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Let the bisector of the angle $C$ in the triangle $ABC$ meet the side $AB$
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at point $P$. We have:
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\[
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\frac{PA}{PB} = \frac{b}{a},
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\]
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where $a$ and $b$ are as in Proposition 1.
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</i>
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<p>
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<b>Proof:</b>
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From $P$ drop perpendicular $PM$ and $PN$ onto the sides $AC$ and $BC$;
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see the diagram below. The right triangles $CMP$ and $CNP$ are congruent
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because they share a common hypotenuse and their angles at $C$ are equal.
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Therefore, $PM = PN$.
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<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="400" height="300">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="The incenter via algebra">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;50,250">
|
||||
<param name="e[2]" value="B;point;fixed;350,250">
|
||||
<param name="e[3]" value="C;point;free;100,50;red;red">
|
||||
<param name="e[4]" value="ABC;polygon;triangle;A,B,C">
|
||||
<param name="e[5]" value="pt1;point;extend;A,C,C,B;none;none">
|
||||
<param name="e[6]" value="P;point;proportion;A,pt1,A,C,A,B,A,B">
|
||||
<param name="e[7]" value="CP;line;connect;C,P;none;none;magenta">
|
||||
<param name="e[8]" value="M;point;foot;P,C,A">
|
||||
<param name="e[9]" value="N;point;foot;P,C,B">
|
||||
<param name="e[10]" value="PM;line;connect;P,M;none;none;green">
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||||
<param name="e[11]" value="PN;line;connect;P,N;none;none;green">
|
||||
<param name="e[12]" value="H;point;foot;C,A,B">
|
||||
<param name="e[13]" value="CH;line;connect;C,H;none;none;cyan">
|
||||
|
||||
<param name="e[14]" value="AH;line;connect;A,H;none;none">
|
||||
<param name="e[15]" value="BN;line;connect;B,N;none;none">
|
||||
<param name="e[16]" value="AM;line;connect;A,M;none;none">
|
||||
|
||||
</applet>
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|
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</td></tr>
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<tr><td>
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<b>
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Drag $C$ to change the geometry.<br>
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Press “r” to reset the diagram to its initial state.<br>
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</b>
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</td></tr></table>
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<p>
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Let $H$ be the foot of the altitude dropped from $C$. Let us note
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the area of the triangle $APC$ may be expressed in two different ways
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in terms of the altitudes $CH$ and $PM$:
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\[
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\frac{1}{2} AP \cdot CH = \frac{1}{2} AC \cdot PM.
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\]
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Similarly, the area of the triangle $BPC$ may be expressed in two
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different way in terms of the altitudes $CH$ and $PN$:
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\[
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\frac{1}{2} BP \cdot CH = \frac{1}{2} BC \cdot PN.
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\]
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Dividing these two equalities and recalling that $PM = PN$,
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we arrive at the desired assertion. <b>QED</b>
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<h2>The proof of Proposition 1</h2>
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<p>
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The diagram below shows the bisector $CC'$ of the angle $C$.
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It is known from elementary geometry that the incenter $O$ lies
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on the bisector.
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From Proposition 2 we have $C'A/C'B = b/a$. It
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follows that $AC'/AB = b/(a+b)$. In terms of the notation of
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<em>the algebra of points</em> introduced earlier in this page,
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this is expressed as:
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\[
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C' = \Big(1 - \frac{b}{a+b} \Big) A + \frac{b}{a+b} B
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||||
= \frac{a}{a+b} A + \frac{b}{a+b} B.
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||||
\]
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Since $O = (1-t)C + tC'$ for some $t$, we arrive at:
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\[
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O = (1-t)C + t\Big[ \frac{a}{a+b} A + \frac{b}{a+b} B \Big].
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||||
\]
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This expresses the triangle's incenter $O$ in terms of its vertices,
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sides lengths, and a yet unknown quantity $t$ that takes values in the range
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0 to 1.
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|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="400" height="300">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="The incenter via algebra">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;100,250">
|
||||
<param name="e[2]" value="B;point;fixed;300,250">
|
||||
<param name="e[3]" value="C;point;free;50,50;red;red">
|
||||
<param name="e[4]" value="ABC;polygon;triangle;A,B,C">
|
||||
<param name="e[5]" value="pt1;point;extend;A,C,C,B;none;none">
|
||||
<param name="e[6]" value="C';point;proportion;A,pt1,A,C,A,B,A,B">
|
||||
<param name="e[7]" value="pt2;point;extend;B,A,A,C;none;none">
|
||||
<param name="e[8]" value="A';point;proportion;B,pt2,B,A,B,C,B,C;none;none">
|
||||
<param name="e[9]" value="pt3;point;extend;C,B,B,A;none;none">
|
||||
<param name="e[10]" value="B';point;proportion;C,pt3,C,B,C,A,C,A;none;none">
|
||||
<param name="e[11]" value="AA';line;connect;A,A';none;none;none">
|
||||
<param name="e[12]" value="BB';line;connect;B,B';none;none;none">
|
||||
<param name="e[13]" value="CC';line;connect;C,C';none;none;magenta">
|
||||
<param name="e[14]" value="O;point;intersection;AA',BB'">
|
||||
<param name="e[15]" value="H;point;foot;O,A,B;none;none">
|
||||
<param name="e[16]" value="IC;circle;radius;O,H;none;none;black;none">
|
||||
|
||||
</applet>
|
||||
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag $C$ to change the geometry.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
</b>
|
||||
</td></tr></table>
|
||||
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<p>
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If we repeat the same calculation by replacing the vertex $C$ by vertex $A$,
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we arrive at:
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\[
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O = (1-s)A + s\Big[ \frac{b}{b+c} B + \frac{c}{b+c} C \Big],
|
||||
\]
|
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where $s$ is yet another unknown that takes values in the range from 0 to 1.
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|
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<p>
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Equating the two expressions for $O$ and collecting the terms, we arrive at:
|
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\[
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\Big[\frac{ta}{a+b} + s - 1\Big] A
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+
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\Big[\frac{tb}{a+b} - \frac{sb}{b+c} \Big] B
|
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+
|
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\Big[1 - t - \frac{sc}{b+c} \Big] C = 0.
|
||||
\]
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|
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In this equations,
|
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if the coefficient of $C$ is nonzero, then we may solve for $C$,
|
||||
obtaining $C = \alpha A + \beta B$ for some $\alpha$
|
||||
and $\beta$. But this would
|
||||
mean that $C$ lies along the line $AB$ which would mean the the
|
||||
three points $A$, $B$ and $C$ are collinear, therefore the
|
||||
triangle $ABC$ is degenerate. We conclude that if the triangle is
|
||||
non-degenerate, then the coefficient of $C$ is zero. Similar
|
||||
arguments show that the coefficients of $A$ and $B$ are zero.
|
||||
Thus, the following system of three equations hold:
|
||||
\[
|
||||
\frac{ta}{a+b} + s - 1 = 0,
|
||||
\quad
|
||||
\frac{tb}{a+b} - \frac{sb}{b+c} = 0,
|
||||
\quad
|
||||
1 - t - \frac{sc}{b+c} = 0.
|
||||
\]
|
||||
Solving this system for the unknowns $t$ and $s$ we obtain:
|
||||
\[
|
||||
t = \frac{a+b}{a+b+c}, \quad s = \frac{b+c}{a+b+c}.
|
||||
\]
|
||||
|
||||
Substituting these in either of the expressions for $O$ we arrive at:
|
||||
\[
|
||||
O = \frac{a}{a+b+C}A + \frac{b}{a+b+C}B + \frac{c}{a+b+C}C,
|
||||
\]
|
||||
which is equivalent to Proposition 1's
|
||||
assertion. <b>QED</b>
|
||||
|
||||
<h2>Extension to tetrahedra</h2>
|
||||
|
||||
<p>
|
||||
Propositions 1 extends to tetrahedra:
|
||||
|
||||
<p>
|
||||
<b>Proposition 3</b>
|
||||
Let
|
||||
$a$, $b$, $c$, $d$
|
||||
be the areas of the faces opposite to the vertices
|
||||
$A$, $B$, $C$, $D$ of the tetrahedron $ABCD$. The the tetrahedron's
|
||||
incenter $O$ is given by:
|
||||
\[
|
||||
O
|
||||
= \frac{a}{\mathcal{A}} A
|
||||
+ \frac{b}{\mathcal{A}} B
|
||||
+ \frac{c}{\mathcal{A}} C
|
||||
+ \frac{d}{\mathcal{A}} D,
|
||||
\]
|
||||
where $\mathcal{A} = a + b + c + d$ is the tetrahedron's surface area.
|
||||
|
||||
<p>
|
||||
This is proved with the aid of the following extension of Proposition 2:
|
||||
|
||||
<p>
|
||||
<b>Proposition 4</b>
|
||||
<i>
|
||||
Let
|
||||
$a$, $b$, $c$, $d$
|
||||
be the areas of the faces opposite to the vertices
|
||||
$A$, $B$, $C$, $D$ of the tetrahedron $ABCD$.
|
||||
Let the bisector plane of the (internal) dihedral angle of edge $CD$
|
||||
intersect edge $AB$ at point $P$.
|
||||
The we have:
|
||||
\[
|
||||
\frac{PA}{PB} = \frac{b}{a}.
|
||||
\]
|
||||
</i>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="400" height="300">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="Triangle's incenter">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;100,250,300">
|
||||
<param name="e[2]" value="B;point;fixed;300,250,300">
|
||||
<param name="e[3]" value="C;point;fixed;350,200,-400">
|
||||
<param name="e[4]" value="D;point;free;200,40;red;red">
|
||||
<param name="e[5]" value="ABCD;polyhedron;tetrahedron;A,B,C,D">
|
||||
<param name="e[6]" value="ABC;plane;3points;A,B,C;none;none;none;none">
|
||||
|
||||
<!-- construct the bisector of the dihedral angle with edge DC -->
|
||||
<param name="e[7]" value="pl1;plane;perpendicular;D,C;none;none;none;none">
|
||||
<param name="e[8]" value="Apl1;point;foot;A,pl1;none;none">
|
||||
<param name="e[9]" value="Bpl1;point;foot;B,pl1;none;none">
|
||||
<param name="e[10]" value="pt1;point;angleBisector;Apl1,D,Bpl1,pl1;none;none">
|
||||
<param name="e[11]" value="tmp1;plane;3points;D,C,pt1;none;none;none;none">
|
||||
<param name="e[12]" value="P;point;intersection;A,B,tmp1">
|
||||
<param name="e[13]" value="DP;line;connect;D,P;none;none;cyan">
|
||||
<param name="e[14]" value="CP;line;connect;C,P;none;none;cyan">
|
||||
|
||||
</applet>
|
||||
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag $D$ to change the geometry.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The plane $CDP$ bisects the dihedral angle of edge $CD$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<p>
|
||||
I leave the proofs to you, the diligent reader. You will find
|
||||
useful information in a
|
||||
<a
|
||||
href="http://groups.google.com/group/sci.math/browse_thread/thread/56c187e018a85111/436f448dfab35a83">discussion
|
||||
in the sci.math newsgroup</a> from January 17, 2006.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on June 26, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html">Geometry Problems and Puzzles</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
|
@ -0,0 +1,156 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>A geometric inequality</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>A geometric inequality</h1>
|
||||
<h4>…and its solution by
|
||||
<a href="mailto:haoyuep@aol.com">Dan Hoey</a></h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="400" height="400">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="A geometric inequality">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;100,300">
|
||||
<param name="e[2]" value="B;point;fixed;300,300">
|
||||
<param name="e[3]" value="ABC;polygon;equilateralTriangle;A,B">
|
||||
<param name="e[4]" value="C;point;vertex;ABC,3">
|
||||
|
||||
<param name="e[5]" value="P;point;free;180,320;red;red">
|
||||
<param name="e[6]" value="PA;line;connect;P,A;none;none;green">
|
||||
<param name="e[7]" value="PB;line;connect;P,B;none;none;green">
|
||||
<param name="e[8]" value="PC;line;connect;P,C;none;none;magenta">
|
||||
|
||||
</applet>
|
||||
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $P$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
Proposition: $PC \le PA + PB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>The problem</h2>
|
||||
|
||||
<p>
|
||||
<b>Proposition:</b> <i>Let $ABC$ be an equilateral triangle and $P$
|
||||
be an arbitrary point in its plane. Then $PC \le PA + PB$.</i>
|
||||
|
||||
<p>
|
||||
This was brought up in
|
||||
<a href="http://mathforum.org/kb/message.jspa?messageID=1086018">a message </a>
|
||||
on the <code>geometry.puzzles</code> newsgroup on November 11, 2001.
|
||||
Go to that message and scroll to the bottom of the page to see the discussion
|
||||
thread.
|
||||
|
||||
<p>
|
||||
On November 22, 2001
|
||||
<a href="mailto:haoyuep@aol.com">Dan Hoey</a>
|
||||
offered a particularly nice solution. He also commented that he
|
||||
had learned that problem may be related to the <em>Van Schooten Theorem</em>,
|
||||
which indeed it is. See
|
||||
<a href="http://www.cut-the-knot.org/Curriculum/Geometry/Pompeiu.shtml">Van
|
||||
Schooten's and Pompeiu's Theorems: What are these?</a> for much detail and
|
||||
historical background.
|
||||
|
||||
<h2>The proof</h2>
|
||||
|
||||
<p>
|
||||
Here is Dan Hoey's proof of the proposition as stated above.
|
||||
|
||||
<p>
|
||||
On the line segment $AP$ construct the equilateral triangle
|
||||
$APD$, as shown in the diagram below, then add the line segment $DC$.
|
||||
|
||||
<p>
|
||||
Let us show that the triangles $APB$ and $ADC$ are congruent. For this,
|
||||
Let us observe that the sides $AP$ and $AB$ in the triangle $APB$
|
||||
equal the sides $AD$ and $AC$ in the triangle $ADC$, by the construction.
|
||||
Moreover, the angles $BAP$ and $CAD$ are equal because each equals
|
||||
the difference of
|
||||
a 60 degree angle and the angle $DAB$. Therefore, the triangles $APB$ and
|
||||
$ADC$ are congruent by the side-angle-side equality.
|
||||
We conclude, in particular, that $PB = DC$.
|
||||
|
||||
<p>
|
||||
In the triangle $PDC$ we have $PC \le PD + DC$. In this inequality
|
||||
replace $PD$ and $DC$ by their equivalents $PA$ and $PB$ to arrive at
|
||||
$PC \le PA + PB$. <b>QED</b>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="400" height="400">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="A geometric inequality">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;100,300">
|
||||
<param name="e[2]" value="B;point;fixed;300,300">
|
||||
<param name="e[3]" value="ABC;polygon;equilateralTriangle;A,B">
|
||||
<param name="e[4]" value="C;point;vertex;ABC,3">
|
||||
|
||||
<param name="e[5]" value="P;point;free;180,320;red;red">
|
||||
<param name="e[6]" value="PA;line;connect;P,A;none;none;green">
|
||||
<param name="e[7]" value="PB;line;connect;P,B;none;none;green">
|
||||
<param name="e[8]" value="PC;line;connect;P,C;none;none;magenta">
|
||||
|
||||
<param name="e[9]" value="APD;polygon;equilateralTriangle;A,P;none;none;none">
|
||||
<param name="e[10]" value="D;point;vertex;APD,3;">
|
||||
<param name="e[11]" value="AD;line;connect;A,D;none;none;lightGray">
|
||||
<param name="e[12]" value="PD;line;connect;P,D;none;none;lightGray">
|
||||
<param name="e[13]" value="CD;line;connect;C,D;none;none;lightGray">
|
||||
|
||||
</applet>
|
||||
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $P$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
Observation: The triangles $APB$ and $ADC$ are congruent.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on November 23, 2001.<br>
|
||||
Cosmetic revisions on June 23, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html">Geometry Problems and Puzzles</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
|
@ -0,0 +1,131 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An equilateral triangle inscribed in a rectangle</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An equilateral triangle inscribed in a rectangle</h1>
|
||||
|
||||
<p>
|
||||
The following solution to the
|
||||
<a href="http://userpages.umbc.edu/~rostamia/Geometry/inscribed-equilateral.html">inscribed triangle puzzle</a>
|
||||
is due to <b>Peter Renz</b>
|
||||
who communicated it to me on December 2016.
|
||||
|
||||
<p>
|
||||
For the sake of completeness, let's begin with the statement of the puzzle.
|
||||
The figure below depicts a equilateral triangle $AEF$ inscribed in a rectangle
|
||||
in such a way that the two share a vertex. We wish to show that the
|
||||
area of the pink triangle ($ECF$) is the sum of the areas of the other two
|
||||
colored triangles.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame000.png" width=500
|
||||
height=417 alt="[frame000.png]">
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As noted in the referring page (see the link above), a solution with
|
||||
the aid of trigonometry is quite straightforward. The purpose of this
|
||||
page is to present a solution in the style of <i>Euclid</i>, without appeal
|
||||
to trigonometry.
|
||||
|
||||
<p>
|
||||
The following animation encapsulates Peter Renz's solution in its
|
||||
entirety.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/waterwheel.gif" width=500
|
||||
height=417 alt="[animation.gif]">
|
||||
</div>
|
||||
|
||||
The animation should be self-explanatory if you stare at it long enough.
|
||||
Nevertheless, I will now proceed to point out the reasoning through
|
||||
several still images extracted from that animation.
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Construct a circle on the diameter $FE$, and then subdivide its boundary
|
||||
into six 60-degree wedges starting at the vertex $C$. Mark the division
|
||||
points $C$, $P$, $Q'$, $C'$, $P'$, and $Q$, as shown.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame007.png" width=500
|
||||
height=417 alt="[frame007.png]">
|
||||
</div>
|
||||
|
||||
<li>
|
||||
Rotate the triangle $FDA$ about $F$ by 60 degrees to bring the edge $FA$ to
|
||||
coincide with $FE$.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame073.png" width=500
|
||||
height=417 alt="[frame073.png]">
|
||||
</div>
|
||||
|
||||
The key observation is that the rotation moves the vertex $D$ into $P'$.
|
||||
|
||||
<li>
|
||||
Further rotate the triangle about the circle's center by 180 degrees to
|
||||
place it in the $FPE$ position.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame111.png" width=500
|
||||
height=417 alt="[frame111.png]">
|
||||
</div>
|
||||
|
||||
<li>
|
||||
Rotate the triangle $EBA$ about $E$ by 60 degrees to bring the edge $EA$ to
|
||||
coincide with $EF$. Then the vertex $B$ will move onto $Q'$ for reasons
|
||||
similar to those explained above.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame177.png" width=500
|
||||
height=417 alt="[frame177.png]">
|
||||
</div>
|
||||
|
||||
<li>
|
||||
The two red line segments in the figure below are parallel and of equal lengths
|
||||
by virtue of being the side and the “radius” of the
|
||||
regular hexagon (not shown) inscribed in the circle.
|
||||
|
||||
<div class="centered">
|
||||
<img src="./inscribed-equilateral-solution/frame180.png" width=500
|
||||
height=417 alt="[frame180.png]">
|
||||
</div>
|
||||
|
||||
The altitudes of the three triangles, dropped from the vertices $C$,
|
||||
$P$, and $Q'$ are shown in dashed lines. It should be clear that the
|
||||
altitude dropped from $C$ is equal in length to the sum of those of
|
||||
the other two altitudes. Since the three triangles share a common base,
|
||||
the area of one is the sum of the areas of the other two. Q.E.D.
|
||||
|
||||
</ol>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html">Geometry Problems and Puzzles</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
|
@ -0,0 +1,135 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An equilateral triangle inscribed in a rectangle</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An equilateral triangle inscribed in a rectangle</h1>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="410" height="370">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An equilateral triangle inscribed in a rectangle">
|
||||
|
||||
<!-- the moving mechanism -->
|
||||
<param name="e[1]" value="O;point;fixed;290,320;0;0;0;0">
|
||||
<param name="e[2]" value="U1;point;fixed;510,320;0;0;0;0">
|
||||
<param name="e[3]" value="V1;point;perpendicular;O,U1;0;0;0;0">
|
||||
<param name="e[4]" value="U;point;angleDivider;U1,O,V1,3;0;0;0;0">
|
||||
<param name="e[5]" value="V;point;angleDivider;V1,O,U1,3;0;0;0;0">
|
||||
<param name="e[6]" value="circ1;circle;radius;O,U;0;0;0;0">
|
||||
<param name="e[7]" value="li1;line;parallel;U,O,U1;0;0;0;0">
|
||||
<param name="e[8]" value="li2;line;parallel;V,O,V1;0;0;0;0">
|
||||
<param name="e[9]" value="W;point;intersection;li1,li2;0;0;0;0">
|
||||
<param name="e[10]" value="VW;line;connect;V,W;0;0;lightGray">
|
||||
<param name="e[11]" value="@;point;lineSegmentSlider;V,W,0,220;red;red">
|
||||
<param name="e[12]" value="li3;line;parallel;@,O,U1;0;0;0;0">
|
||||
<param name="e[13]" value="li4;line;chord;circ1,li3;0;0;0;0">
|
||||
<param name="e[14]" value="X1;point;first;li4;0;0;0;0">
|
||||
|
||||
<!-- the triangle -->
|
||||
<param name="e[15]" value="A;point;fixed;50,320">
|
||||
<param name="e[16]" value="V2;point;perpendicular;A,U1;0;0;0;0">
|
||||
<param name="e[17]" value="li5;line;parallel;A,O,X1;0;0;0;0">
|
||||
<param name="e[18]" value="X2;point;last;li5;0;0;0;0">
|
||||
<param name="e[19]" value="X;point;extend;A,X2,A,X2;0;0;0;0">
|
||||
|
||||
<param name="e[20]" value="tri1;polygon;equilateralTriangle;X,A;0;0;0;0">
|
||||
<param name="e[21]" value="Y;point;vertex;tri1,3;0;0;0;0">
|
||||
<param name="e[22]" value="B;point;midpoint;X,Y">
|
||||
<param name="e[23]" value="ABC;polygon;equilateralTriangle;A,B">
|
||||
<param name="e[24]" value="C;point;vertex;ABC,3">
|
||||
|
||||
<!-- the rectangle -->
|
||||
<param name="e[25]" value="D;point;foot;B,A,U1">
|
||||
<param name="e[26]" value="F;point;foot;C,A,V2">
|
||||
<param name="e[27]" value="FE;line;parallel;F,A,D;0;0;0;0">
|
||||
<param name="e[28]" value="E;point;last;FE">
|
||||
<param name="e[29]" value="rect;polygon;quadrilateral;A,D,E,F;0;0;black;0">
|
||||
<param name="e[30]" value="ADB;polygon;triangle;A,D,B;0;0;0;pink">
|
||||
<param name="e[31]" value="ACF;polygon;triangle;A,C,F;0;0;0;pink">
|
||||
<param name="e[32]" value="BCE;polygon;triangle;B,C,E;0;0;0;cyan">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Slide the “@” up and down to change the geometry.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
Proposition: The blue area equals the sum of the two pink areas.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Problem statement</h2>
|
||||
|
||||
<p>
|
||||
The diagram above shows an equilateral triangle inscribed in a rectangle
|
||||
in such a way that the two have a vertex in common. This subdivides the
|
||||
rectangle into four disjoint triangles.
|
||||
The original equilateral triangle is shown in white
|
||||
in the diagram; the other three are shown in color.
|
||||
|
||||
<p>
|
||||
<b>Proposition</b>
|
||||
<em>
|
||||
The area of the blue triangle equals the sum
|
||||
of the areas of the two pink triangles.
|
||||
</em>
|
||||
|
||||
<p>
|
||||
The trigonometric proof is quite straightforward. I don't
|
||||
know of a classical proof <i>a la</i> <span class="name">Euclid</span>.
|
||||
(Well, actually I haven't tried much.)
|
||||
If you can think of a neat non-trigonometric proof, let me know. I will
|
||||
put it here with due credit.
|
||||
|
||||
<p>
|
||||
This problem appeared as a conjecture
|
||||
<a href="http://mathforum.org/kb/thread.jspa?forumID=129&messageID=1083967">in an article</a>
|
||||
in the <code>geometry.puzzles</code> newsgroup on March 15, 1997.
|
||||
|
||||
<p>
|
||||
<b>Note added January 8, 2017:</b>
|
||||
Here is a
|
||||
<a href="inscribed-equilateral-solution.html">clever solution</a>
|
||||
that <b>Peter Renz</b> sent me a in December 2016. Thanks, Peter!
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on July 2, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html">Geometry Problems and Puzzles</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
|
@ -0,0 +1,142 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>Triangles with common base</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>Triangles with common base</h1>
|
||||
<h4>An interesting problem proposed by
|
||||
<a href="mailto:stevebg@adelphia.net">Steve Gray</a></h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="650" height="250">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="Triangles with common base">
|
||||
|
||||
<param name="e[1]" value="A;point;fixed;50,215">
|
||||
<param name="e[2]" value="B;point;fixed;250,215">
|
||||
<param name="e[3]" value="T0;polygon;equilateralTriangle;A,B">
|
||||
<param name="e[4]" value="C;point;vertex;T0,3">
|
||||
|
||||
<param name="e[5]" value="D;point;free;115,173;red;red">
|
||||
<param name="e[6]" value="DC;line;connect;D,C;none;none;orange">
|
||||
<param name="e[7]" value="DA;line;connect;D,A;none;none;blue">
|
||||
<param name="e[8]" value="DB;line;connect;D,B;none;none;green">
|
||||
|
||||
<param name="e[9]" value="P;point;fixed;300,215">
|
||||
<param name="e[10]" value="Q;point;fixed;600,215">
|
||||
<param name="e[11]" value="PQ;line;connect;P,Q">
|
||||
|
||||
<param name="e[12]" value="B';point;similar;P,Q,C,A,D">
|
||||
<param name="e[13]" value="PB';line;connect;P,B';none;none;green">
|
||||
<param name="e[14]" value="QB';line;connect;Q,B';none;none;green">
|
||||
|
||||
<param name="e[15]" value="C';point;similar;P,Q,A,B,D">
|
||||
<param name="e[16]" value="PC';line;connect;P,C';none;none;orange">
|
||||
<param name="e[17]" value="QC';line;connect;Q,C';none;none;orange">
|
||||
|
||||
<param name="e[18]" value="A';point;similar;P,Q,B,C,D">
|
||||
<param name="e[19]" value="PA';line;connect;P,A';none;none;blue">
|
||||
<param name="e[20]" value="QA';line;connect;Q,A';none;none;blue">
|
||||
<param name="e[21]" value="T1;polygon;triangle;B',C',A';none;none;lightGray">
|
||||
|
||||
<!-- construct the centroid -->
|
||||
<param name="e[22]" value="midA'B';point;midpoint;A',B';none;none">
|
||||
<param name="e[23]" value="li1;line;connect;C',midA'B';none;none;none">
|
||||
<param name="e[24]" value="midB'C';point;midpoint;B',C';none;none">
|
||||
<param name="e[25]" value="li2;line;connect;A',midB'C';none;none;none">
|
||||
<param name="e[26]" value="M;point;intersection;li1,li2">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $D$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
Proposition: $A'B'C'$ is equilateral and its centroid $M$ is fixed.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>The construction</h2>
|
||||
|
||||
<p>
|
||||
This problem was proposed by
|
||||
<a href="mailto:stevebg@adelphia.net">Steve Gray</a>
|
||||
in the <code>geometry.puzzles</code> newsgroup
|
||||
(<a href="http://mathforum.org/kb/message.jspa?messageID=1088552">see
|
||||
the original message</a>) on July 26, 2002.
|
||||
Scroll to the bottom of that page for a link to the solution.
|
||||
|
||||
<p>
|
||||
Consider an equilateral triangle $ABC$, a line segment $PQ$,
|
||||
and an arbitrary point $D$, as seen in the diagram above.
|
||||
On the segment $PQ$ construct three triangles
|
||||
$PC'Q$,
|
||||
$PA'Q$,
|
||||
$PB'Q$,
|
||||
similar to the triangles
|
||||
$ADB$,
|
||||
$BDC$,
|
||||
$CDA$,
|
||||
respectively.
|
||||
|
||||
<p>
|
||||
<b>Proposition 1:</b>
|
||||
The triangle $A'B'C'$ is equilateral.
|
||||
|
||||
<p>
|
||||
<b>Proposition 2:</b>
|
||||
The centroid of $A'B'C'$ is independent of $D$.
|
||||
|
||||
|
||||
<p>
|
||||
Steve adds:
|
||||
<blockquote>
|
||||
<p>
|
||||
Now generalize this for a regular $n$-gon. The new points form a
|
||||
regular $n$-gon whose centroid is independent of $D$.
|
||||
This problem is original so far as I know. I am interested in the
|
||||
simplest synthetic solution; no algebra, please.
|
||||
</blockquote>
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
July 26, 2002.<br>
|
||||
Cosmetic revisions on June 17, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html">Geometry Problems and Puzzles</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
|
@ -0,0 +1,403 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>Construction by Chris Alberts</h4>
|
||||
|
||||
<p>
|
||||
The construction described in this page is due to Chris Alberts, who
|
||||
sent it to me in an email on March 15, 2011.
|
||||
I have paraphrased and rearranged his construction, but the
|
||||
differences from the original are cosmetic. The error analysis is mine.
|
||||
|
||||
<p>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="810" height="600">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
<!-- <param name="align" value="left"> -->
|
||||
<param name="pivot" value="O">
|
||||
|
||||
<!-- The angle AOB -->
|
||||
<param name="e[1]" value="O;point;fixed;400,400">
|
||||
<param name="e[2]" value="A;point;fixed;790,400">
|
||||
<param name="e[3]" value="c;circle;radius;O,A;none;none;lightGray;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;c,720,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
|
||||
<!-- The inner circle -->
|
||||
<param name="e[7]" value="A';point;fixed;530,400">
|
||||
<param name="e[8]" value="c';circle;radius;O,A';none;none;lightGray;none">
|
||||
<param name="e[9]" value="l1;line;chord;OB,c';none;none;none">
|
||||
<param name="e[10]" value="B';point;first;l1">
|
||||
|
||||
<!-- the points P, R, M -->
|
||||
<param name="e[11]" value="E;point;midpoint;B',O">
|
||||
<param name="e[12]" value="c'';circle;radius;B',O;none;none;00aa00;none">
|
||||
<param name="e[13]" value="l2;line;parallel;E,A,O;none;none;none">
|
||||
<param name="e[14]" value="l3;line;chord;l2,c'';none;none;none">
|
||||
<param name="e[15]" value="P;point;first;l3">
|
||||
<param name="e[16]" value="l4;line;chord;l2,c';none;none;none">
|
||||
<param name="e[17]" value="R;point;last;l4">
|
||||
<param name="e[18]" value="l8;line;connect;E,R">
|
||||
<param name="e[19]" value="M;point;midpoint;P,R">
|
||||
|
||||
<!-- the points N and F -->
|
||||
<param name="e[20]" value="l5;line;chord;B',M,c';none;none;none">
|
||||
<param name="e[21]" value="N;point;last;l5">
|
||||
<param name="e[22]" value="l9;line;connect;B',N;none;none;none;none"> <!-- -->
|
||||
<param name="e[23]" value="l6;line;parallel;B',A,O;none;none;none">
|
||||
<param name="e[24]" value="c1;circle;radius;N,B';none;none;none;none">
|
||||
<param name="e[25]" value="l7;line;chord;l6,c1;none;none;none">
|
||||
<param name="e[26]" value="F;point;last;l7">
|
||||
|
||||
<!-- the point G -->
|
||||
<param name="e[27]" value="T1;polygon;triangle;B',N,F;none;none;lightGray;ddffcc">
|
||||
<param name="e[28]" value="l10;line;chord;N,F,c;none;none;none">
|
||||
<param name="e[29]" value="G;point;first;l10">
|
||||
<param name="e[30]" value="l11;line;connect;F,G;none;none;cyan">
|
||||
<param name="e[31]" value="l12;line;chord;G,O,c';none;none;none">
|
||||
<param name="e[32]" value="H;point;last;l12">
|
||||
<param name="e[33]" value="l13;line;connect;G,H;none;none;lightGray">
|
||||
|
||||
<!-- the points H, D, and J -->
|
||||
<param name="e[34]" value="D;point;last;l1">
|
||||
<param name="e[35]" value="l14;line;connect;O,D">
|
||||
<param name="e[36]" value="l15;line;parallel;D,O,A;none;none;none">
|
||||
<param name="e[37]" value="c2;circle;radius;H,D;none;none;none;none">
|
||||
<param name="e[38]" value="l16;line;chord;l15,c2;none;none;none">
|
||||
<param name="e[39]" value="J;point;last;l16">
|
||||
<param name="e[40]" value="T2;polygon;triangle;D,H,J;none;none;lightGray;ddffcc">
|
||||
|
||||
<!-- the point U -->
|
||||
<param name="e[41]" value="l17;line;chord;H,J,c;none;none;none">
|
||||
<param name="e[42]" value="K;point;first;l17">
|
||||
<param name="e[43]" value="l18;line;connect;J,K;none;none;cyan">
|
||||
<param name="e[44]" value="c3;circle;radius;A,K;none;none;none;none">
|
||||
<param name="e[45]" value="l19;line;bichord;c,c3;none;none;none">
|
||||
<param name="e[46]" value="T;point;last;l19">
|
||||
<param name="e[47]" value="l20;line;connect;O,T;none;none;red">
|
||||
<param name="e[48]" value="l21;line;connect;K,T;none;none;lightGray">
|
||||
|
||||
<!-- labels for the circles -->
|
||||
<param name="e[49]" value="C'';point;circleSlider;c'',150,0;00aa00;none">
|
||||
<param name="e[50]" value="C';point;fixed;255,420;lightGray;none">
|
||||
<param name="e[51]" value="C;point;fixed;120,120;lightGray;none">
|
||||
|
||||
<!-- angle markers -->
|
||||
<param name="e[52]" value="po1;point;fixed;445,400;none;none">
|
||||
<param name="e[53]" value="ci1;circle;radius;O,po1;none;none;none;none">
|
||||
<param name="e[54]" value="li1;line;chord;O,T,ci1;none;none;none">
|
||||
<param name="e[55]" value="po2;point;first;li1;none;none">
|
||||
<param name="e[56]" value="li2;line;chord;O,B,ci1;none;none;none">
|
||||
<param name="e[57]" value="po3;point;first;li2;none;none">
|
||||
<param name="e[58]" value="se1;sector;sector;O,po1,po2;none;none;black;yellow">
|
||||
<param name="e[59]" value="se2;sector;sector;O,po2,po3;none;none;black;orange">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$
|
||||
(but keep $AOB$ to less than 90 degrees).<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line ($OT$) is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<p>
|
||||
Consider the circular arc $AB$ on the circle $C$ centered at $O$,
|
||||
shown in the diagram above.
|
||||
Assume that the angle $AOB$ is between 0 and 90 degrees.
|
||||
To trisect $AOB$, do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Draw the circle $C'$ centered at $O$ with a radius 1/3 $OA$.
|
||||
Mark $A'$ and $B'$ its intersections with the line segments $OA$ and $OB$,
|
||||
respectively.
|
||||
|
||||
<li>
|
||||
Draw the circle $C''$ (shown in green) centered at $B'$ through the point $O$.
|
||||
|
||||
<li>
|
||||
Let $E$ be the midpoint of the line segment $OB'$.
|
||||
Draw a line through $E$ parallel to $OA$ and mark its intersections
|
||||
$P$ and $R$ with the circles $C''$ and $C'$, as shown.
|
||||
|
||||
<li>
|
||||
Let $M$ be the midpoint of the line segment $PR$. Draw the line $B'M$
|
||||
and extend to the intersection point $N$ with the circle $C'$.
|
||||
|
||||
<li>
|
||||
Draw a line through $B'$ parallel to $OA$, and select the point $F$ on
|
||||
it so that $NB' = NF$.
|
||||
|
||||
<li>
|
||||
Extend the line segment $NF$ to intersect the circle $C$ at $G$.
|
||||
|
||||
<li>
|
||||
Draw the line $GO$ and extend it to the point of intersection $H$
|
||||
with the circle $C'$.
|
||||
<em>Note:</em> If you look at the diagram closely, you will be able to
|
||||
see that the line segments $GO$ and $GN$
|
||||
are <em>not</em> collinear.
|
||||
|
||||
<li>
|
||||
Let $D$ be diagonally opposite the point $B'$ in the
|
||||
circle $C'$. Draw a line through $D$ parallel to $OA$, and
|
||||
select the point $J$ on it so that $HD=HJ$.
|
||||
<em>Note:</em> Although it is impossible to discern visually,
|
||||
the line segments $OH$ and $HJ$
|
||||
are <em>not</em> collinear.
|
||||
|
||||
<li>
|
||||
Extend the line segment $HJ$ to intersect the circle $C$ at $K$.
|
||||
|
||||
<li>
|
||||
Reflect $K$ about the line $OA$ to get the point $T$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error analysis</h2>
|
||||
|
||||
<p>
|
||||
The calculation of the coordinates of all the points that appear
|
||||
in the construction is elementary but
|
||||
the resulting expressions are massively large, therefore I will refrain
|
||||
from putting them on this web page. (I did the calculations in <em>Maple</em>).
|
||||
<p>
|
||||
Let $\alpha$ and $\beta(\alpha)$ be the sizes of the angles $AOB$ and $AOT$,
|
||||
respectively.
|
||||
Express the trisection error as $e(\alpha) = \frac{\alpha}{3} - \beta(\alpha)$.
|
||||
It turns out that $e(0) = e(\pi/2) = 0$.
|
||||
In the range $0$ to $\pi/2$ the error is the largest near
|
||||
1.22175 radians = 70.0013 degrees. The maximum error is
|
||||
$2.32\times10^{-18}$ radians = $1.33\times10^{-16}$ degrees.
|
||||
|
||||
<p>
|
||||
Expanding $\beta(\alpha)$ in power series we get:
|
||||
\[
|
||||
\beta(\alpha)
|
||||
= \frac{1}{3} \alpha +
|
||||
\frac{5^9}{2^{13} \cdot 3^{40}} \alpha^{27} + O(\alpha^{29})
|
||||
= \frac{1}{3} \alpha +
|
||||
\frac{1953125}{99595595440594360737792} \alpha^{27} + O(\alpha^{29}).
|
||||
\]
|
||||
|
||||
|
||||
<h2>Alberts' refinement</h2>
|
||||
|
||||
<p>
|
||||
The extraordinary precision of Chris Alberts' trisection
|
||||
is a result of the application of a refinement technique which
|
||||
I will call <em>Alberts' refinement</em>.
|
||||
The 10 steps of his trisection procedure, described above,
|
||||
consist of three distinct stages:
|
||||
<ol>
|
||||
<li><em>Stage 1,</em> corresponding to the steps 1–5 of the construction,
|
||||
produces the angle $FB'N$ which is roughly one third of the angle $AOB$.
|
||||
The trisection error at this stage may be as large as 0.7 degrees.
|
||||
|
||||
<li><em>Stage 2,</em> corresponding to the steps 6–7 of the construction,
|
||||
produces the angle $A'OH$ which is quite close to $\frac13 \alpha$. The worst
|
||||
error is 0.00013791 degrees and
|
||||
occurs when $AOB$ is near 70 degrees.
|
||||
|
||||
<li><em>Stage 3,</em> corresponding to the steps 8–10 of the construction,
|
||||
produces the angle $AOT$ which, as noted above, is within
|
||||
$1.33\times10^{-16}$ degrees of the exact trisection.
|
||||
</ol>
|
||||
|
||||
Each of the stages 2 and 3 consists of one application
|
||||
of <em>Alberts' refinement</em> which may be formulated as
|
||||
a stand-alone geometric proposition illustrated
|
||||
in the diagram below.
|
||||
|
||||
<p>
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="450" height="320">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="Alberts' refinement">
|
||||
|
||||
<!-- the lines L2, L1, L3 -->
|
||||
<param name="e[1]" value="O;point;fixed;40,250">
|
||||
<param name="e[2]" value="B';point;fixed;380,250;none;none">
|
||||
<param name="e[3]" value="L2;line;connect;O,B';black">
|
||||
<param name="e[4]" value="c1;circle;radius;O,B';none;none;none;none">
|
||||
<param name="e[5]" value="A';point;circleSlider;c1,100,10;none;none">
|
||||
<param name="e[6]" value="L1;line;connect;O,A';black">
|
||||
<param name="e[7]" value="s1;sector;sector;O,B',A';none;none;blue;none">
|
||||
<param name="e[8]" value="X;point;circleSlider;c1,450,110;red;red">
|
||||
<param name="e[9]" value="L3;line;connect;O,X;none;none;blue">
|
||||
|
||||
<!-- the points A, P, B -->
|
||||
<param name="e[10]" value="p1;point;fixed;160,250;none;none">
|
||||
<param name="e[11]" value="A;point;cutoff;L1,O,p1">
|
||||
<param name="e[12]" value="c2;circle;radius;A,O;none;none;none;none">
|
||||
<param name="e[13]" value="l1;line;chord;O,X,c2">
|
||||
<param name="e[14]" value="P;point;last;l1">
|
||||
<param name="e[15]" value="T1;polygon;triangle;O,A,P;none;none;black;ffddee">
|
||||
<param name="e[16]" value="c3;circle;radius;P,O;none;none;none;none">
|
||||
<param name="e[17]" value="l2;line;chord;O,B',c3">
|
||||
<param name="e[18]" value="B;point;last;l2">
|
||||
<param name="e[19]" value="T2;polygon;triangle;O,P,B;none;none;black;ddffcc">
|
||||
|
||||
<!-- the circle of radius 3 -->
|
||||
<param name="e[20]" value="l3;line;extend;O,A,O,A;none;none;none;none">
|
||||
<param name="e[21]" value="l4;line;extend;l3,O,A;none;none;none;none">
|
||||
<param name="e[22]" value="p2;point;last;l4;none;none">
|
||||
<param name="e[23]" value="c4;circle;radius;A,O,p2;none;none;magenta;none">
|
||||
|
||||
<!-- the point G -->
|
||||
<param name="e[24]" value="l5;line;chord;P,B,c4;none;none;none">
|
||||
<param name="e[25]" value="G;point;first;l5">
|
||||
<param name="e[26]" value="l6;line;connect;B,G;none;none;cyan">
|
||||
<param name="e[27]" value="l7;line;connect;A,G">
|
||||
|
||||
<!-- the horizontal line -->
|
||||
<param name="e[28]" value="l8;line;parallel;A,O,p1">
|
||||
<param name="e[29]" value="L;point;last;l8">
|
||||
|
||||
<!-- angle markers -->
|
||||
<param name="e[30]" value="p3;point;fixed;80,250;none;none">
|
||||
<param name="e[31]" value="c5;circle;radius;O,p3;none;none;none;none">
|
||||
<param name="e[32]" value="l9;line;chord;O,P,c5;none;none;none">
|
||||
<param name="e[33]" value="p4;point;first;l9;none;none">
|
||||
<param name="e[34]" value="s2;sector;sector;O,p3,p4;none;none;none;blue">
|
||||
|
||||
<param name="e[35]" value="p5;point;cutoff;l8,O,p3;none;none">
|
||||
<param name="e[36]" value="c6;circle;radius;A,p5;none;none;none;none">
|
||||
<param name="e[37]" value="l10;line;chord;A,G,c6;none;none;none">
|
||||
<param name="e[38]" value="p6;point;first;l10;none;none">
|
||||
<param name="e[39]" value="s3;sector;sector;A,p6,p5;none;none;none;red">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $X$.
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
If the blue angle is an approximate trisection of the angle $AOB$,
|
||||
then the red angle is a much better trisection.<br>
|
||||
Note that the red angle hardly changes as the point $X$ varies.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<p>
|
||||
The lines $L_1$ and $L_2$ intersect at the point $O$. Suppose that the
|
||||
line $OX$ is a crude trisector of the angle between $L_1$ and $L_2$.
|
||||
The rest of the diagram shows a
|
||||
straightedge and compass construction
|
||||
that produces a much finer trisection.
|
||||
|
||||
Here are the details of the construction:
|
||||
|
||||
<ol>
|
||||
<li> Pick an arbitrary point $A$ (other than $O$) on the line $L_1$.
|
||||
<li> Locate the point $P$ on the line $OX$ so that $AO = AP$.
|
||||
<li> Locate the point $B$ on the line $L_2$ so that $PO = PB$.
|
||||
<li> Draw a circle centered at $A$ and radius equal to three
|
||||
times the length of $OA$. (This circle is shown in magenta.)
|
||||
<li> Extend the line segment $PB$ to a intersect that circle at a point $G$.
|
||||
(This extension is shown in cyan.)
|
||||
<li> Draw a line $AL$ through $A$ and parallel to the line $L_2$.
|
||||
</ol>
|
||||
|
||||
The angle $LAG$ is quite close to being one third of the angle $AOB$.
|
||||
To see this, let us write $\alpha$, $\beta$, and $\beta'$ for
|
||||
the measures (in the radian units)
|
||||
of the angles $AOB$, $XOB$, and $LAG$, respectively.
|
||||
Let $\beta = \frac13 \alpha + \delta$ and $\beta' = \frac13 \alpha + \delta'$.
|
||||
Thus, $\delta$ and $\delta'$ measure the trisection errors corresponding to the
|
||||
angles $\beta$ and $\beta'$.
|
||||
|
||||
<p>
|
||||
A quite straightforward calculation, involving an application of the
|
||||
<em>law of sines</em> in the triangle $APG$ leads to the equation:
|
||||
\[
|
||||
\delta' = \delta - \arcsin\Big(\frac13 \sin3\delta\Big).
|
||||
\]
|
||||
Expanding this into power series in $\delta$, we obtain:
|
||||
\[
|
||||
\delta' = \frac43 \delta^3 - \frac45 \delta^7 + O(\delta^9).
|
||||
\]
|
||||
This explains the notable efficiency of the refinement. For instance,
|
||||
if the value of $\beta$ has two significant digits after the decimal point,
|
||||
the value of $\beta'$ will have six significant digits after the decimal
|
||||
point.
|
||||
|
||||
<p>
|
||||
<i>Remark 1:</i> Move the point $X$ in the diagram and note
|
||||
how insensitive the angle
|
||||
$LAG$ is to the choice of $X$. This indicates that even a crude
|
||||
initial approximation produces an excellent trisection.
|
||||
|
||||
<p>
|
||||
<i>Remark 2:</i> If you examine closely Chris Alberts' trisection
|
||||
described earlier in
|
||||
this page, you will find buried in it two instances of <em>Alberts'
|
||||
refinement</em>.
|
||||
|
||||
|
||||
<h2>A final comment</h2>
|
||||
|
||||
<p>
|
||||
Comparing the precision of the trisection described in this
|
||||
page to those of others presented on my website may not
|
||||
seem to be quite fair. After all, <em>any</em> approximate trisection may
|
||||
be applied iteratively to refine its own result.
|
||||
Nevertheless, I am making an exception in this case because in
|
||||
Chris Alberts' trisection, the iterative refinement is an
|
||||
inherent feature of the method.
|
||||
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a> on
|
||||
March 23, 2011.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,268 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>Construction by
|
||||
<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a></h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="570" height="360">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<!-- the angle AOB -->
|
||||
<param name="e[1]" value="O;point;fixed;260,310">
|
||||
<param name="e[2]" value="A;point;fixed;520,310">
|
||||
<param name="e[3]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[5]" value="B;point;circleSlider;C,20,0;red;red">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
|
||||
|
||||
<!-- point P quadrisects the arc AB -->
|
||||
<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
|
||||
<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
|
||||
<param name="e[11]" value="P;point;first;Ot2">
|
||||
<param name="e[12]" value="AP;line;connect;A,P;none;none;green">
|
||||
|
||||
<!-- the arc A'OB' -->
|
||||
<param name="e[13]" value="A';point;fixed;455,310">
|
||||
<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
|
||||
<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
|
||||
<param name="e[16]" value="B';point;first;t9">
|
||||
<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
|
||||
|
||||
<!-- P' -->
|
||||
<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
|
||||
<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
|
||||
<param name="e[20]" value="P';point;first;t4">
|
||||
<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
|
||||
<param name="e[22]" value="s2;point;first;s1;none;none">
|
||||
<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
|
||||
<param name="e[24]" value="A'P';line;connect;A',P';none;none;green">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[25]" value="p1;point;fixed;290,310;none;none">
|
||||
<param name="e[26]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[27]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[28]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[29]" value="l2;line;chord;s3,c1;none;none;none">
|
||||
<param name="e[30]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[31]" value="S1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
<param name="e[32]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[33]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[34]" value="S2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<H2>The Basic Construction</H2>
|
||||
|
||||
<p>
|
||||
Here is a very simple straightedge and compass
|
||||
construction of an approximate angle trisector due to
|
||||
<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a>.
|
||||
|
||||
<p>
|
||||
Let us represent the angle by
|
||||
the circular arc $AB$ centered at $O$; see the diagram above.
|
||||
The angle's size may be anything from 0 to 180 degrees.
|
||||
To trisect, do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Quadrisect the angle $AOB$, that is, divide it into four
|
||||
equal parts. The arc $AP$ in the diagram above represents one quarter
|
||||
of the original arc $AB$. Let $L$ be the length of the chord $AP$ (shown in green).
|
||||
|
||||
<li>
|
||||
Draw a circular arc (shown in orange)
|
||||
centered at $O$ and radius 3/4 of $OA$. Mark $A'$ and $B'$ its intersections
|
||||
with the rays $OA$ and $OB$, respectively.
|
||||
|
||||
<li>
|
||||
Swing an arc (not shown) of radius $L$ centered at $A'$ and mark $P'$ its intersection with
|
||||
the arc $A'B'$, as shown.
|
||||
|
||||
</ol>
|
||||
|
||||
<p>
|
||||
The line $OP'$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta=\tau(\alpha)$ be the sizes of the angles $AOB$ and $A'OP'$,
|
||||
respectively. It is straightforward to show that
|
||||
\[
|
||||
\beta
|
||||
= 2 \arcsin\big(\frac{4}{3}\sin\frac{\alpha}{8}\big)
|
||||
= \frac{\alpha}{3} + \frac{7}{2^7\cdot3^4}\alpha^3 + O(\alpha^5)
|
||||
= \frac{\alpha}{3} + \frac{7}{10368}\alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
<!-- The first two terms of the series are the same as those
|
||||
in trisect-dunham.html. The third terms are different.
|
||||
b_baker := 2*arcsin(4/3*sin(a/8));
|
||||
series(b_baker,a);
|
||||
b_durham := a/2 - arctan(sin(a/4 - arcsin(sin(a/4)/2))*4/3);
|
||||
series(b_durham, a);
|
||||
-->
|
||||
|
||||
<p>
|
||||
The error
|
||||
$
|
||||
\ds e(\alpha) = \tau(\alpha) - \frac{\alpha}{3}
|
||||
$
|
||||
is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2)$ = 0.002695 radians = 0.154 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.0237 radians = 1.360 degrees.
|
||||
|
||||
|
||||
<h2>Iterative Improvement</h2>
|
||||
|
||||
<p>
|
||||
As we see in the asymptotic expansion shown above, the
|
||||
angle $\tau(\alpha)$ is slightly larger than the target value of $\alpha/3$.
|
||||
Making three copies of the constructed angle, and putting them
|
||||
end-to-end as in arcs $A'P'$, $P'P''$, and $P''P'''$ shown in the diagram below,
|
||||
we arrive at the endpoint $P'''$ which is very slightly off the point $B'$,
|
||||
and just outside the arc $A'B'$. The constructible angle $B'OP'''$ is exactly
|
||||
three times the error $e(\alpha)$.
|
||||
If we were able to trisect $B'OP'''$ exactly, then we
|
||||
would know the error, and consequently will have achieved
|
||||
the exact trisection of the original angle.
|
||||
Of course the exact trisection of $B'OP'''$ is impossible in general, but we
|
||||
may repeat the method outlined in the <em>Basic Construction</em> above
|
||||
to obtain an <em>approximate</em> trisection of $B'OP'''$,
|
||||
which will yield $ \tau\big(3\tau(\alpha) - \alpha\big) $,
|
||||
and consequently an improved trisection $\tau_{\mathrm{improved}}(\alpha)$
|
||||
of the original angle:
|
||||
\[
|
||||
\tau_{\mathrm{improved}}(\alpha) = \tau(\alpha) - \tau\big(3\tau(\alpha) - \alpha\big)
|
||||
= \frac{\alpha}{3} - \frac{7^4}{2^{28}\cdot3^{13}} \alpha^9 +
|
||||
O(\alpha^{11}).
|
||||
\]
|
||||
|
||||
The error
|
||||
$ \ds e_{\mathrm{improved}}(\alpha) = \frac{\alpha}{3} - \tau_{\mathrm{improved}}(\alpha)$
|
||||
is monotonically increasing in $\alpha$. In particular,
|
||||
$e_{\mathrm{improved}}(\pi/2) = 1.5\times 10^{-9}$ radians
|
||||
$ = 8.6\times10^{-8}$ degrees.
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="570" height="360">
|
||||
<param name=background value="ffffff">
|
||||
<param name=title value="An angle trisection">
|
||||
|
||||
<!-- the angle AOB -->
|
||||
<param name="e[1]" value="O;point;fixed;260,310">
|
||||
<param name="e[2]" value="A;point;fixed;520,310">
|
||||
<param name="e[3]" value="OA;line;connect;O,A">
|
||||
<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[5]" value="B;point;circleSlider;C,50,0">
|
||||
<param name="e[6]" value="OB;line;connect;O,B">
|
||||
<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
|
||||
|
||||
<!-- point P quadrisects the arc AB -->
|
||||
<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
|
||||
<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
|
||||
<param name="e[11]" value="P;point;first;Ot2">
|
||||
<param name="e[12]" value="OP;line;connect;O,P;none;none;none">
|
||||
|
||||
<!-- the arc A'OB' -->
|
||||
<param name="e[13]" value="A';point;fixed;455,310">
|
||||
<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
|
||||
<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
|
||||
<param name="e[16]" value="B';point;first;t9">
|
||||
<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
|
||||
|
||||
<!-- P' -->
|
||||
<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
|
||||
<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
|
||||
<param name="e[20]" value="P';point;first;t4">
|
||||
<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
|
||||
<param name="e[22]" value="s2;point;first;s1;none;none">
|
||||
<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
|
||||
|
||||
<!-- P'' -->
|
||||
<param name="e[24]" value="t5;circle;radius;P',A,P;none;none;none;none">
|
||||
<param name="e[25]" value="t6;line;bichord;t5,C';none;none;none">
|
||||
<param name="e[26]" value="P'';point;first;t6">
|
||||
|
||||
<!-- P'''
|
||||
Note the trailing spaces after P'''. These become a part of the label!
|
||||
-->
|
||||
<param name="e[27]" value="t7;circle;radius;P'',A,P;none;none;none;none">
|
||||
<param name="e[28]" value="t8;line;bichord;t7,C';none;none;none">
|
||||
<param name="e[29]" value="P''' ;point;first;t8">
|
||||
<param name="e[30]" value="OP''';line;connect;O,P''' ;none;none;black">
|
||||
<param name="e[31]" value="u1;sector;sector;O,B',P''' ;none;none;none;magenta">
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line is an approximate trisector of the angle $AOB$.<br>
|
||||
The arcs $P'P''$ and $P''P'''$ are copies of $A'P'$. The endpoint $P'''$
|
||||
is just slightly off the point $B'$.<br>
|
||||
The (very small and nearly indiscernible)
|
||||
angle $B'OP'''$ is three times the trisection error.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a> on
|
||||
May 31, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,154 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>
|
||||
Construction by <a href="http://cdsmith.wordpress.com/2009/04/20/old-memories-about-trisecting-angles/">cdsmith</a>
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="600" height="450">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;250,340">
|
||||
<param name="e[2]" value="A;point;fixed;450,340">
|
||||
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;cir1,400,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
|
||||
<param name="e[7]" value="AB;line;connect;A,B;none;none;lightGray">
|
||||
<param name="e[8]" value="M;point;midpoint;A,B">
|
||||
<param name="e[9]" value="cA;circle;radius;A,M;none;none;green;none">
|
||||
<param name="e[10]" value="cM;circle;radius;M,A;none;none;green;none">
|
||||
<param name="e[11]" value="cB;circle;radius;B,M;none;none;green;none">
|
||||
<param name="e[12]" value="li1;line;bichord;cA,cM;none;none;none">
|
||||
<param name="e[13]" value="C;point;first;li1">
|
||||
<param name="e[14]" value="li2;line;bichord;cM,cB;none;none;none">
|
||||
<param name="e[15]" value="D;point;first;li2">
|
||||
|
||||
<param name="e[16]" value="OC;line;connect;O,C;none;none;red">
|
||||
<param name="e[17]" value="OD;line;connect;O,D;none;none;red">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[18]" value="p1;point;fixed;285,340;none;none">
|
||||
<param name="e[19]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[20]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[21]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[22]" value="l2;line;chord;O,C,c1;none;none;none">
|
||||
<param name="e[23]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[24]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
<param name="e[25]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[26]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[27]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
To trisect the angle $AOB$ (with $OA=OB$), do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Find the midpoint $M$ of the line segment $AB$.
|
||||
|
||||
<li>
|
||||
Draw circles centered at $A$, $M$, and $B$, each of radius $\frac{1}{2}AB$,
|
||||
and mark their intersection points $C$ and $D$, as shown in the diagram
|
||||
above.
|
||||
|
||||
</ol>
|
||||
|
||||
The lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOC$,
|
||||
respectively. One may verify that
|
||||
\[
|
||||
\beta =
|
||||
\arctan\bigg(
|
||||
\frac{
|
||||
\sin\frac{\alpha}{2}
|
||||
\sin\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
|
||||
}{
|
||||
1 +
|
||||
\sin\frac{\alpha}{2}
|
||||
\cos\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
|
||||
}
|
||||
\bigg)
|
||||
=
|
||||
\frac{1}{4}\alpha + \frac{\sqrt{3}}{16} \alpha^2 - \frac{1}{16} \alpha^3
|
||||
+ O(\alpha^4).
|
||||
\]
|
||||
|
||||
<p>
|
||||
This says that $\ds \beta \approx \frac{1}{4}\alpha$ when $\alpha$ is small,
|
||||
so small angles are quadrisected, rather than trisected!
|
||||
(This is clearly visible in the interactive diagram above.)
|
||||
For not-so-small angles,
|
||||
the method works reasonably well. In fact, it produces
|
||||
<em>exact trisection</em> for angles $\alpha=\pi/2$ and $\alpha=\pi$.
|
||||
|
||||
<p>
|
||||
The worst error in
|
||||
the range $0 \le \alpha \le \pi$ is
|
||||
0.0214 radians = 1.23 degrees.
|
||||
This occurs at
|
||||
$\alpha=2\arctan(\sqrt{3}\pm\sqrt{2})$ which corresponds to
|
||||
angles of approximately 35 degrees and 145 degrees.
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on June 14, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,169 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>From page 26 of<br>
|
||||
Underwood Dudley, <i>The Trisectors</i>, 2nd edition, 1996.
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="600" height="250">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;50,200">
|
||||
<param name="e[2]" value="A;point;fixed;200,200">
|
||||
<param name="e[3]" value="C;point;extend;O,A,O,A">
|
||||
<param name="e[4]" value="D;point;extend;O,C,O,A">
|
||||
<param name="e[5]" value="AD;line;connect;A,D;none;none;lightGray">
|
||||
<param name="e[6]" value="OA;line;connect;O,A;none;none;blue">
|
||||
|
||||
<param name="e[7]" value="c0;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[8]" value="B;point;circleSlider;c0,70,100;red;red">
|
||||
<param name="e[9]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[10]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
|
||||
|
||||
<param name="e[11]" value="L;line;parallel;B,O,D;none;none;black">
|
||||
|
||||
<param name="e[12]" value="circDE;circle;radius;C,D;none;none;none;none">
|
||||
<param name="e[13]" value="x8;line;chord;L,circDE;none;none;none">
|
||||
<param name="e[14]" value="E;point;last;x8">
|
||||
<param name="e[15]" value="CE;line;connect;C,E;none;none;lightGray">
|
||||
<param name="e[16]" value="arcDE;sector;sector;C,D,E;none;none;lightGray;none">
|
||||
|
||||
<param name="e[17]" value="F;point;foot;E,AD">
|
||||
<param name="e[18]" value="EF;line;connect;E,F;none;none;lightGray">
|
||||
|
||||
<param name="e[19]" value="circFT;circle;radius;O,F;none;none;none;none">
|
||||
<param name="e[20]" value="x13;line;chord;L,circFT;none;none;none">
|
||||
<param name="e[21]" value="T;point;first;x13">
|
||||
<param name="e[22]" value="OT;line;connect;O,T;none;none;red">
|
||||
|
||||
<param name="e[23]" value="arcFT;sector;sector;O,F,T;none;none;green;none">
|
||||
|
||||
<param name="e[24]" value="circOA;circle;radius;O,A;none;none;none">
|
||||
<param name="e[25]" value="x2;line;chord;OT,circOA;none;none;none">
|
||||
<param name="e[26]" value="P;point;first;x2;none;none;none">
|
||||
|
||||
<param name="e[27]" value="p1;point;fixed;80,200;none;none">
|
||||
<param name="e[28]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[29]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[30]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[31]" value="l2;line;chord;OT,c1;none;none;none">
|
||||
<param name="e[32]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[33]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<param name="e[34]" value="p2;point;fixed;80,200;none;none">
|
||||
<param name="e[35]" value="c2;circle;radius;O,p2;none;none;none;none">
|
||||
<param name="e[36]" value="l3;line;chord;OT,c2;none;none;none">
|
||||
<param name="e[37]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[38]" value="l4;line;chord;OB,c2;none;none;none">
|
||||
<param name="e[39]" value="q4;point;first;l4;none;none">
|
||||
<param name="e[40]" value="s2;sector;sector;O,q3,q4;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$
|
||||
(but keep it less than 90 degrees).<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line, $OT$, is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
We wish to trisect the given angle $AOB$. Assume the angle is less than
|
||||
90 degrees; see the diagram above.
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Draw a line through $B$ parallel to $OA$.
|
||||
|
||||
<li>
|
||||
Extend $OA$ and mark off $AC$ and $CD$ along it, each equal to $OA$.
|
||||
|
||||
<li>
|
||||
Draw the arc $DE$ with center $C$ and radius $CD$.
|
||||
|
||||
<li>
|
||||
Drop a perpendicular from $E$ to $OD$ and let $F$ be the foot of the perpendicular.
|
||||
|
||||
<li>
|
||||
Draw the arc $FT$ with center $O$ and radius $OF$ (shown in green).
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $FOT$,
|
||||
respectively. It is straightforward to show that
|
||||
\[
|
||||
\beta
|
||||
= \frac{\sin \alpha}{2+\cos \alpha}
|
||||
= \frac{\alpha}{3} - \frac{1}{2^2 \cdot 3^3 \cdot 5} \alpha^5 +
|
||||
O(\alpha^7)
|
||||
= \frac{\alpha}{3} - \frac{1}{540} \alpha^5 + O(\alpha^7).
|
||||
\]
|
||||
The error
|
||||
$
|
||||
\ds e(\alpha) = \frac{\alpha}{3} - \beta
|
||||
$
|
||||
is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2) =$ 0.0236 radians = 1.352 degrees.
|
||||
|
||||
<p>
|
||||
It is interesting that the error is $O(\alpha^5)$ rather than $O(\alpha^3)$
|
||||
as one might have expected. Despite this, the method's accuracy
|
||||
is not particularly remarkable for angles that are not very close to zero.
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on July 26, 2002.<br>
|
||||
Cosmetic revisions on June 6, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,155 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>Construction attributed to A. G. O<br>
|
||||
From page 133 of<br>
|
||||
Underwood Dudley, <i>The Trisectors</i>, 2nd edition, 1996.
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="450" height="260">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;210,225">
|
||||
<param name="e[2]" value="A;point;fixed;420,225">
|
||||
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;cir1,0,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[7]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
|
||||
<param name="e[8]" value="pt1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[9]" value="C;point;cutoff;O,pt1,O,A">
|
||||
<param name="e[10]" value="OC;line;connect;O,C;none;none;lightGray">
|
||||
|
||||
<param name="e[11]" value="D;point;fixed;280,225">
|
||||
<param name="e[12]" value="E;point;cutoff;OB,O,D">
|
||||
<param name="e[13]" value="G;point;cutoff;OC,O,D">
|
||||
<param name="e[14]" value="F;point;extend;A,O,O,D">
|
||||
<param name="e[15]" value="OF;line;connect;O,F;none;none;lightGray">
|
||||
<param name="e[16]" value="sec1;sector;sector;O,D,E;none;none;orange">
|
||||
<param name="e[17]" value="sec2;sector;sector;O,E,F;none;none;lightGray">
|
||||
|
||||
<param name="e[18]" value="li1;line;chord;F,G,cir1;none;none;none">
|
||||
<param name="e[19]" value="T;point;last;li1">
|
||||
<param name="e[20]" value="FT;line;connect;F,T;none;none;lightGray">
|
||||
<param name="e[21]" value="OT;line;connect;O,T;none;none;red">
|
||||
|
||||
<!-- angle markers -->
|
||||
<param name="e[22]" value="p1;point;fixed;240,225;none;none">
|
||||
<param name="e[23]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[24]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[25]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[26]" value="l2;line;chord;OT,c1;none;none;none">
|
||||
<param name="e[27]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[28]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<param name="e[29]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[30]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[31]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line, $OT$, is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
We wish to trisect the given angle $AOB$ represented by the circular arc
|
||||
$AB$ centered at $O$, as shown in the diagram above.
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Draw the bisector $OC$ of the angle $AOB$.
|
||||
|
||||
<li>
|
||||
Draw the circular arc $DE$ centered at $O$ so that $OD = \frac{1}{3} OA$.
|
||||
Let $G$ be where the line $OC$ intersects the arc $DE$.
|
||||
|
||||
<li>
|
||||
Locate $F$ on the extension of $OA$ so that $OF=OD$.
|
||||
|
||||
<li>
|
||||
Connect $FG$ and extend to the intersection point $T$ with
|
||||
the arc $AB$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ (shown in red) is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$,
|
||||
respectively. It is straightforward to show that
|
||||
\[
|
||||
\beta
|
||||
= \frac{\alpha}{4} + \arcsin\Big(
|
||||
\frac{1}{3}\sin\frac{1}{4}\alpha \Big)
|
||||
= \frac{1}{3}\alpha - \frac{1}{2^4\cdot3^4} \alpha^3 + O(\alpha^7)
|
||||
= \frac{1}{3}\alpha - \frac{1}{1296} \alpha^3 + O(\alpha^7).
|
||||
\]
|
||||
The term after $\alpha^3$ is $\alpha^7$. That's not a typo.
|
||||
|
||||
<p>
|
||||
The error
|
||||
$
|
||||
\ds e(\alpha) = \frac{\alpha}{3} - \beta
|
||||
$
|
||||
is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2) =$ 0.003 radians = 0.171 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.024 radians = 1.367 degrees.
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on July 26, 2002.<br>
|
||||
Cosmetic revisions on June 13, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,267 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>
|
||||
R. L. Durham, <i>A simple construction for the approximate trisection
|
||||
of an angle</i>,
|
||||
American Mathematical Monthly,
|
||||
vol. 51, no. 4, April 1944, pp. 217–218.
|
||||
</h4>
|
||||
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="450" height="350">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;210,300">
|
||||
<param name="e[2]" value="A;point;fixed;420,300">
|
||||
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;cir1,0,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[7]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
|
||||
<param name="e[8]" value="pt1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[9]" value="C;point;cutoff;O,pt1,O,A">
|
||||
<param name="e[10]" value="OC;line;connect;O,C;none;none;lightGray">
|
||||
|
||||
<!-- O-pt2 is 1/3 of OA -->
|
||||
<param name="e[11]" value="pt2;point;fixed;280,300;none;none">
|
||||
<param name="e[12]" value="F;point;proportion;O,A,O,pt2,C,A,C,A">
|
||||
<param name="e[13]" value="G;point;extend;A,C,F,C">
|
||||
<param name="e[14]" value="AG;line;connect;A,G;none;none;lightGray">
|
||||
<param name="e[15]" value="cir2;circle;radius;G,F;none;none;green;none">
|
||||
|
||||
<param name="e[16]" value="li1;line;perpendicular;C,O,O,A;none;none;lightGray">
|
||||
<param name="e[17]" value="li2;line;chord;li1,cir2;none;none;none">
|
||||
<param name="e[18]" value="T;point;first;li2">;
|
||||
<param name="e[19]" value="OT;line;connect;O,T;none;none;red">;
|
||||
|
||||
<!-- angle markers -->
|
||||
<param name="e[20]" value="p1;point;fixed;240,300;none;none">
|
||||
<param name="e[21]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[22]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[23]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[24]" value="l2;line;chord;OT,c1;none;none;none">
|
||||
<param name="e[25]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[26]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<param name="e[27]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[28]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[29]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line, $OT$, is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
We wish to trisect the given angle $AOB$ represented by the circular arc
|
||||
$AB$ centered at $O$, as shown in the diagram above.
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Draw the bisector $OC$ of the angle $AOB$.
|
||||
|
||||
<li>
|
||||
Draw the chord $AC$ and trisect it at point $F$ so that $CF=\frac{1}{3}CA$.
|
||||
|
||||
<li>
|
||||
Locate point $G$ on the extension of the chord $AC$ so that $GC=CF$.
|
||||
|
||||
<li>
|
||||
Draw a circle (shown in green) centered at $G$ and through point $F$.
|
||||
|
||||
<li>
|
||||
Let $T$ on the green circle be such that $TC$ is perpendicular to $OC$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ (shown in red) is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<p>
|
||||
<span class="name">R. L. Durham</span>
|
||||
(see the reference at the top of this page) goes one step further.
|
||||
Using the line $OT$ as a starting point, he produces a substantially
|
||||
better approximation $OT'$. The construction for this second stage is
|
||||
complex and not particularly pretty, so I won't go into that here.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta $ be the sizes of the angles $AOB$ and $AOT$,
|
||||
respectively. It is possible to show that
|
||||
\[
|
||||
\beta
|
||||
= \frac{1}{2}{\alpha}
|
||||
-
|
||||
\arctan\bigg(
|
||||
\frac{4}{3}\sin\Big(
|
||||
\frac{\alpha}{4} -
|
||||
\arcsin\big(\frac{1}{2}\sin\frac{\alpha}{4}\big)
|
||||
\Big)\bigg)
|
||||
= \frac{1}{3}\alpha + \frac{7}{2^7\cdot3^4} \alpha^3 + O(\alpha^5)
|
||||
= \frac{1}{3}\alpha + \frac{7}{10368} \alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
<!-- The first two terms of the series are the same as those
|
||||
in trisect-baker.html. The third terms are different.
|
||||
b_baker := 2*arcsin(4/3*sin(a/8));
|
||||
series(b_baker,a);
|
||||
b_durham := a/2 - arctan(sin(a/4 - arcsin(sin(a/4)/2))*4/3);
|
||||
series(b_durham, a);
|
||||
-->
|
||||
|
||||
|
||||
<p>
|
||||
The error $\ds e(\alpha) = \beta - \frac{\alpha}{3}$
|
||||
is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2) =$ 0.00265 radians = 0.152 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.0218 radians = 1.252 degrees.
|
||||
|
||||
<h2>Calculation details</h2>
|
||||
|
||||
<p>
|
||||
The derivation of the formula for $\beta$ shown above is a straightforward
|
||||
application of trigonometry. Here are the details.
|
||||
|
||||
<p>
|
||||
The angles $AOC$ and $ACT$ subtend the arc $AB$ of the circle centered
|
||||
at $O$. Since $AOC = \alpha/2$, then $ACT=\alpha/4$. Consequently,
|
||||
the angle $GCT$ is $\pi - \alpha/4$.
|
||||
|
||||
<p>
|
||||
We apply the law of sines in the triangle $GCT$. (The edge $GT$
|
||||
is not shown in the diagram to reduce clutter.) Let us write
|
||||
$\theta$ for the angle $CGT$. The the angle $CTG$ is
|
||||
$\alpha/4 - \theta$. The law of sines is:
|
||||
\[
|
||||
\frac{\sin\theta}{CT} = \frac{\sin(\alpha/4-\theta)}{CG} = \frac{\sin(\pi-\alpha/4)}{GT}.
|
||||
\]
|
||||
But $GT=2CG$, therefore $2\sin(\alpha/4-\theta) = \sin(\pi-\alpha/4) = \sin(\alpha/4)$, whence
|
||||
\[
|
||||
\theta = \frac{\alpha}{4} - \arcsin\big(\frac{1}{2}\sin\frac{\alpha}{4}\big).
|
||||
\]
|
||||
Going back to the equation of law of since, we compute $CT$:
|
||||
\[
|
||||
CT = \frac{\sin\theta}{\sin(\alpha/4)} GT.
|
||||
\]
|
||||
But $GT = GF = 2CF = \frac{2}{3} AC
|
||||
= \frac{2}{3} \big(2OA\sin\frac{\alpha}{4}\big) = \frac{4}{3} OA\sin\frac{\alpha}{4}$.
|
||||
We conclude that $CT = \frac{4}{3} OA\sin\theta$.
|
||||
Then $COT = \arctan \frac{CT}{OC} = \arctan \frac{4}{3} \sin\theta$. Finally, the angle
|
||||
$\beta$, which equals $COA - COT$, is given by:
|
||||
\[
|
||||
\beta = \frac{\alpha}{2} - \arctan \frac{4}{3} \sin\theta.
|
||||
\]
|
||||
Substituting the expression for $\theta$ calculated above, we arrive at the desired
|
||||
expression for $\beta$.
|
||||
|
||||
<h2>Aside</h2>
|
||||
|
||||
<p>
|
||||
In 1990, the well-known logician
|
||||
<span class="name">Willard Van Orman Quine</span>
|
||||
wrote an expository article in
|
||||
the <em>Mathematics Magazine</em> where he proves that
|
||||
some angles cannot be trisected by ruler and compass. The proof
|
||||
is elementary (but not short) and requires
|
||||
nothing but simple algebra. Here is the full reference:
|
||||
|
||||
<h4>
|
||||
W. V. Quine, <i>Elementary proof that some angles cannot be trisected by ruler
|
||||
and compass</i>,
|
||||
Mathematics Magazine,
|
||||
vol. 63, no. 2, April 1990, pp. 95–105.
|
||||
</h4>
|
||||
|
||||
<p>
|
||||
In a prefatory note he refers to <span class="name">Durham</span>, the author of the trisection
|
||||
described in this web page. He writes:
|
||||
|
||||
<blockquote>
|
||||
<p>
|
||||
This purely expository paper dates from April 1946.
|
||||
<span class="name">Robert Lee Durham</span>,
|
||||
president emeritus of Southern Seminary Junior and College,
|
||||
had sent me a hundred dollars and asked me to make it clear to him
|
||||
why an angle cannot in general be trisected by ruler
|
||||
and compass. He had himself presented a way of almost trisecting
|
||||
any angle by ruler and compass, to an accuracy for acute angles of
|
||||
1/720 of a degree. [Here he refers to <span class="name">Durham</span>'s
|
||||
1944 article cited at the top of this web page.]
|
||||
|
||||
<p>
|
||||
I welcomed the money and the occasion to familiarize myself
|
||||
with the famous proof. I was guided in large part by
|
||||
<span class="name">L. E. Dickson</span>,
|
||||
<em>Why it is impossible trisect to an
|
||||
angle or construct a regular polygon of 7 or 9 sides by ruler and compass,</em>
|
||||
Mathematics Teacher, vol. 14 (1921), 217–223.
|
||||
|
||||
<p>
|
||||
<span class="name">Mr. Durham</span>
|
||||
expressed satisfaction with my report and proposed
|
||||
paying for publishing it as a pamphlet. With his approval I
|
||||
submitted instead to a mathematics journal. After waiting
|
||||
nineteen months for a decision from the journal, I recalled
|
||||
the paper and dropped the matter.
|
||||
</blockquote>
|
||||
|
||||
<p>
|
||||
<span class="name">Quine</span>
|
||||
goes on to explain how this article was eventually published
|
||||
more than 50 years after it was written.
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on June 13, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,185 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>
|
||||
Free Jamison, <i>Trisection Approximation</i>, American Mathematical Monthly,
|
||||
vol. 61, no. 5, May 1954, pp. 334–336.
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="700" height="400">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;200,200">
|
||||
<param name="e[2]" value="A;point;fixed;200,350">
|
||||
<param name="e[3]" value="C1;circle;radius;O,A;none;none;lightGray;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;C1,280,0;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
|
||||
<!-- the points F, D, C -->
|
||||
<param name="e[7]" value="x1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[8]" value="x2;point;angleBisector;x1,O,B;none;none">
|
||||
<param name="e[9]" value="x3;point;angleBisector;x1,O,x2;none;none">
|
||||
<param name="e[10]" value="F;point;cutoff;O,x2,O,A">
|
||||
<param name="e[11]" value="D;point;cutoff;O,x3,O,A">
|
||||
<param name="e[12]" value="C;point;extend;F,O,F,O">
|
||||
|
||||
<!-- the lines FC, CE, OE -->
|
||||
<param name="e[13]" value="FC;line;connect;F,C;none;none;lightGray">
|
||||
<param name="e[14]" value="CD;line;connect;C,D;none;none;green">
|
||||
<param name="e[15]" value="E;point;extend;C,D,C,F">
|
||||
<param name="e[16]" value="DE;line;connect;D,E;none;none;green">
|
||||
<param name="e[17]" value="OE;line;connect;O,E;none;none;red">
|
||||
|
||||
<param name="e[18]" value="p1;point;fixed;225,200;none;none">
|
||||
<param name="e[19]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[20]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[21]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[22]" value="l2;line;chord;OE,c1;none;none;none">
|
||||
<param name="e[23]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[24]" value="s1;sector;sector;O,q1,q2;none;none;black;yellow">
|
||||
|
||||
<param name="e[25]" value="p2;point;fixed;225,200;none;none">
|
||||
<param name="e[26]" value="c2;circle;radius;O,p2;none;none;none;none">
|
||||
<param name="e[27]" value="l3;line;chord;OE,c2;none;none;none">
|
||||
<param name="e[28]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[29]" value="l4;line;chord;OB,c2;none;none;none">
|
||||
<param name="e[30]" value="q4;point;first;l4;none;none">
|
||||
<param name="e[31]" value="s2;sector;sector;O,q3,q4;none;none;black;orange">
|
||||
|
||||
<!-- needed for the error analysis, not the construction -->
|
||||
<param name="e[32]" value="OD;line;connect;O,D;none;none;lightGray">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$
|
||||
(but stay on the right half of the circle).<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line $OE$ is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
|
||||
<h2>The construction</h2>
|
||||
|
||||
<p>
|
||||
This construction, due to Free Jamison
|
||||
(see the reference at the top of this page)
|
||||
is a more accurate variant of the construction described in
|
||||
<a href="trisect-jamison.html">a simpler construction</a>.
|
||||
|
||||
<p>
|
||||
Consider the circular arc $AB$ centered at $O$, shown in the diagram above.
|
||||
Assume the angle $AOB$ is between 0 and 180 degrees.
|
||||
To trisect $AOB$, do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li> Pick the points $F$ and $D$ on the arc $BA$ such that
|
||||
arc $BF$ = 2/8 of the arc $BA$ and
|
||||
arc $BD$ = 3/8 of the arc $BA$.
|
||||
|
||||
<li> Extend $FO$ to intersect the circle at a point $C$.
|
||||
|
||||
<li> Draw the line $CD$ and extend it to a point $E$ such that $DE$ equals the
|
||||
circle's diameter.
|
||||
|
||||
</ol>
|
||||
|
||||
<p>
|
||||
The line $OE$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $EOB$, respectively.
|
||||
The angle $FOD$ equals $\alpha/8$ by the construction, therefore the
|
||||
angle $FCD$, which is half the central angle $FOD$, is equal to
|
||||
$\alpha/16$.
|
||||
The triangle $DOC$ is isosceles, therefore the angle $ODC$ also equals $\alpha/16$.
|
||||
|
||||
<p>
|
||||
In the triangle $OED$, let $x$ and $y$ be the sizes of the angles
|
||||
$OED$ and $EOD$, respectively. Since the sum $x+y$ of the triangle's internal
|
||||
angles equals the triangle's
|
||||
external angle $ODC$, we have $x+y = \alpha/16$. Let us note, however,
|
||||
that the angle $y$ equals $DOB$ minus $EOB$. Thus $y = 3\alpha/8 - \beta$,
|
||||
whence $x = \beta - 5\alpha/16$.
|
||||
|
||||
<p>
|
||||
In the triangle $OED$, the side $DE$ is twice the side $OD$ by the construction,
|
||||
therefore the law of sines gives $\sin y = 2 \sin x$. Consequently,
|
||||
$\sin(3\alpha/8 - \beta) = 2 \sin(\beta - 5\alpha/16)$.
|
||||
Solving this for $\beta$ we arrive at:
|
||||
\[
|
||||
\beta
|
||||
= \frac{5}{16} \alpha + \arctan \frac{\sin(a/16)}{2+\cos(a/16)}
|
||||
= \frac{1}{3} \alpha - \frac{1}{2^{12}\cdot3^4} \alpha^3 + O(\alpha^5)
|
||||
= \frac{1}{3} \alpha - \frac{1}{331776} \alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
|
||||
<p>
|
||||
We see that the trisection error $e(\alpha) = \alpha/3 - \beta$ is given by:
|
||||
\[
|
||||
e(\alpha) = \frac{1}{48}\alpha - \arctan \frac{\sin(a/16)}{2+\cos(a/16)}.
|
||||
\]
|
||||
(This formula is also given in Jamison's article.)
|
||||
The function $e(a)$ is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2)$ = 0.0000117 radians = 0.00067 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.000093756 radians = 0.00537 degrees.
|
||||
Quite impressive!
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a> on
|
||||
July 22, 2002.
|
||||
<br>Cosmetic revisions on June 7, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,219 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>
|
||||
Construction attributed to C. R. Lindberg, as reported in<br>
|
||||
Free Jamison, <i>Trisection Approximation</i>, American Mathematical Monthly,
|
||||
vol. 61, no. 5, May 1954, pp. 334–336.
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="700" height="400">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;200,200">
|
||||
<param name="e[2]" value="A;point;fixed;200,350">
|
||||
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|
||||
<param name="e[4]" value="B;point;circleSlider;C1,350,0;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[7]" value="Dtmp;point;midpoint;A,B;none;none">
|
||||
<param name="e[8]" value="D;point;cutoff;O,Dtmp,O,A">
|
||||
<param name="e[9]" value="OD;line;connect;O,D;none;none;lightGray">
|
||||
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|
||||
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|
||||
<param name="e[12]" value="CD;line;connect;C,D;none;none;green">
|
||||
<param name="e[13]" value="E;point;extend;C,D,B,C">
|
||||
<param name="e[14]" value="DE;line;connect;D,E;none;none;green">
|
||||
<param name="e[15]" value="OE;line;connect;O,E;none;none;red">
|
||||
<param name="e[16]" value="E';point;cutoff;O,E,O,A;none;none">
|
||||
|
||||
<param name="e[17]" value="p1;point;fixed;200,225;none;none">
|
||||
<param name="e[18]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[19]" value="l1;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[20]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[21]" value="l2;line;chord;OE,c1;none;none;none">
|
||||
<param name="e[22]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[23]" value="s1;sector;sector;O,q2,q1;none;none;black;orange">
|
||||
|
||||
<param name="e[24]" value="p2;point;fixed;200,225;none;none">
|
||||
<param name="e[25]" value="c2;circle;radius;O,p2;none;none;none;none">
|
||||
<param name="e[26]" value="l3;line;chord;OA,c2;none;none;none">
|
||||
<param name="e[27]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[28]" value="l4;line;chord;OE,c2;none;none;none">
|
||||
<param name="e[29]" value="q4;point;first;l4;none;none">
|
||||
<param name="e[30]" value="s2;sector;sector;O,q3,q4;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$
|
||||
(but stay on the right half of the circle).<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line $OE$ is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>The construction</h2>
|
||||
|
||||
<p>
|
||||
According to Jamison (see the reference at the top of this page)
|
||||
the construction's main idea
|
||||
comes from an unpublished work by C. R. Lindberg.
|
||||
|
||||
<p>
|
||||
Consider the circular arc $AB$ centered at $O$, shown in the diagram above.
|
||||
Assume the angle $AOB$ is between 0 and 180 degrees.
|
||||
To trisect $AOB$, do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Extend $BO$ to intersect the circle at a point $C$.
|
||||
|
||||
<li>
|
||||
Draw the bisector $OD$ of the angle $AOB$.
|
||||
|
||||
<li>
|
||||
Draw the line $CD$ and extend it to a point $E$ such that $DE$ equals the
|
||||
circle's diameter.
|
||||
|
||||
</ol>
|
||||
|
||||
<p>
|
||||
The line $OE$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<p>
|
||||
Here is a heuristic explanation for why the construction works,
|
||||
as explained by Jamison. The key lies
|
||||
in the observation that (i) in the triangle $ODE$ the angles $O$ and $E$ are
|
||||
“small”, and (ii) the side $ED$ is twice as long as the side $OD$.
|
||||
Therefore from the law of sines we have $\sin(O)/\sin(E) = ED/OD = 2$,
|
||||
which implies that the angle $O$ is approximately twice the angle $E$ in
|
||||
the triangle $ODE$.
|
||||
|
||||
<p>
|
||||
Let the measure of the angle $OED$ be $x$.
|
||||
Then the triangle's external angle at $D$, that is the angle $ODC$, is the sum of
|
||||
the internal angles $O$ and $E$, therefore it is approximately $3x$.
|
||||
Therefore the angle $OCD$ is $3x$. Therefore the angle $BOD$ is $6x$.
|
||||
Since the angle $E'OD$ is $2x$, we conclude that the angle $BOE'$ is $4x$.
|
||||
Furthermore, since the angle $BOD$ is $6x$, the angle $BOA$ is $12x$. This shows
|
||||
that $BOA$ is 3 times $BOE'$, as asserted.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $EOB$, respectively.
|
||||
The angle $DOB$ is half of $AOB$ by the construction, therefore it is equal to
|
||||
$\alpha/2$. Consequently the angle $DCB$, which is half the
|
||||
central angle $DOB$, equals $\alpha/4$.
|
||||
The triangle $DOC$ is isosceles, therefore the angle $ODC$ also equals $\alpha/4$.
|
||||
|
||||
<p>
|
||||
In the triangle $OED$, let $x$ and $y$ be the sizes of the angles
|
||||
$OED$ and $EOD$, respectively. Since the sum $x+y$ of the triangle's internal
|
||||
angles equals the triangle's
|
||||
external angle $ODC$, we have $x+y = \alpha/4$. Let us note, however,
|
||||
that the angle $y$ equals $DOB$ minus $EOB$. Thus $y = \alpha/2 - \beta$,
|
||||
whence $x = \beta - \alpha/4$.
|
||||
|
||||
<p>
|
||||
In the triangle $OED$, the side $DE$ is twice the side $OD$ by the construction,
|
||||
therefore the law of sines gives $\sin y = 2 \sin x$. Consequently,
|
||||
$\sin(\alpha/2 - \beta) = 2 \sin(\beta - \alpha/4)$. Solving this for $\beta$
|
||||
we arrive at:
|
||||
\[
|
||||
\beta
|
||||
= \frac{1}{4} \alpha + \arctan \frac{\sin(a/4)}{2+\cos(a/4)}
|
||||
= \frac{1}{3} \alpha - \frac{1}{2^6\cdot3^4} \alpha^3 + O(\alpha^5)
|
||||
= \frac{1}{3} \alpha - \frac{1}{5184} \alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
|
||||
<p>
|
||||
We see that the trisection error $e(\alpha) = \alpha/3 - \beta$ is given by:
|
||||
\[
|
||||
e(\alpha) = \frac{1}{12}\alpha - \arctan \frac{\sin(a/4)}{2+\cos(a/4)}.
|
||||
\]
|
||||
(This formula is also given in Jamison's article.)
|
||||
The function $e(a)$ is monotonically increasing.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2)$ = 0.000757 radians = 0.0434 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.0063 radians = 0.361 degrees.
|
||||
That's quite good for such a simple construction.
|
||||
|
||||
<h2>An interesting coincidence</h2>
|
||||
|
||||
<p>
|
||||
The angle $\beta$ constructed by this method coincides <em>exactly</em>
|
||||
with that of <a href="trisect-pllana.html">Pllana's construction</a>,
|
||||
where $\beta$ is given as:
|
||||
\[
|
||||
\beta
|
||||
= \arctan \frac
|
||||
{\sin\frac{\alpha}{2} + 2\sin\frac{\alpha}{4}}
|
||||
{\cos\frac{\alpha}{2} + 2\cos\frac{\alpha}{4}}.
|
||||
\]
|
||||
One way to verify that the seemingly different expressions
|
||||
for $\beta$ are in fact identical,
|
||||
is to compare their derivatives. In both cases we have:
|
||||
\[
|
||||
\frac{d\beta}{d\alpha} =
|
||||
\frac{3(1 + \cos\frac{\alpha}{4})}{2(5 + 4\cos\frac{\alpha}{4})}.
|
||||
\]
|
||||
|
||||
<h2>An extension</h2>
|
||||
|
||||
<p>
|
||||
Although this construction is quite good as is, Jamison proceeds to give
|
||||
<a href="trisect-jamison-ext.html">an extension of Lindberg's method</a>
|
||||
which requires a bit more work but is substantially more accurate.
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a> on
|
||||
July 22, 2002.<br>
|
||||
Cosmetic revisions on June 7, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,173 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>Construction due to
|
||||
<a href="mailto:avniu66@hotmail.com">Avni Pllana</a></h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="450" height="260">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;210,225">
|
||||
<param name="e[2]" value="A;point;fixed;410,225">
|
||||
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;cir1,0,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[7]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
|
||||
<param name="e[8]" value="pt1;point;angleBisector;A,O,B;none;none">
|
||||
<param name="e[9]" value="C;point;cutoff;O,pt1,O,A">
|
||||
<param name="e[10]" value="li1;line;connect;O,C;none;none;lightGray">
|
||||
<param name="e[11]" value="M;point;midpoint;O,C">
|
||||
<param name="e[12]" value="pt3;point;angleBisector;A,O,C;none;none">
|
||||
<param name="e[13]" value="D;point;cutoff;O,pt3,O,A">
|
||||
<param name="e[14]" value="li2;line;connect;O,D;none;none;lightGray">
|
||||
<param name="e[15]" value="N;point;midpoint;M,D">
|
||||
<param name="e[16]" value="MD;line;connect;M,D;none;none;cyan">
|
||||
<param name="e[17]" value="li3;line;cutoff;O,N,O,A;none;none;red">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[18]" value="p1;point;fixed;240,225;none;none">
|
||||
<param name="e[19]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[20]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[21]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[22]" value="l2;line;chord;O,N,c1;none;none;none">
|
||||
<param name="e[23]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[24]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[25]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[26]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
<param name="e[27]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
This approximate trisection, due to Avni Pllana, was announced
|
||||
<a href="http://mathforum.org/kb/message.jspa?messageID=1084688">in a
|
||||
message</a>
|
||||
in the <code>geometry.puzzles</code> newsgroup on July 23, 2003.
|
||||
Scroll to the bottom of that page to view the related discussion thread.
|
||||
|
||||
<p>
|
||||
Consider the angle $AOB$ given by the circular arc $AB$ centered at $O$,
|
||||
as shown in the diagram above.
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Pick points $C$ and $D$ on the arc $AB$ so that $OC$ bisects the angle $AOB$
|
||||
and $OD$ bisects the angle $AOC$.
|
||||
|
||||
<li>
|
||||
Let $M$ be the midpoint of the line segment $OC$.
|
||||
|
||||
<li>
|
||||
Let $N$ be the midpoint of the line segment $MD$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $ON$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AON$,
|
||||
respectively. One may verify that:
|
||||
\[
|
||||
\beta
|
||||
= \arctan \frac
|
||||
{\sin\frac{\alpha}{2} + 2\sin\frac{\alpha}{4}}
|
||||
{\cos\frac{\alpha}{2} + 2\cos\frac{\alpha}{4}}
|
||||
= \frac{1}{3}\alpha - \frac{1}{2^6\cdot3^4} \alpha^3 + O(\alpha^5)
|
||||
= \frac{1}{3}\alpha - \frac{1}{5184} \alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
|
||||
<em>Hint:</em> Represent the points as complex numbers
|
||||
in the polar form $re^{i\theta}$.
|
||||
|
||||
<p>
|
||||
The error
|
||||
$
|
||||
\ds e(\alpha) = \frac{\alpha}{3} - \beta
|
||||
$
|
||||
increases monotonically with $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2)$ = 0.000757 radians = 0.0434 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.00630 radians = 0.361 degrees.
|
||||
That's quite good for such a simple construction.
|
||||
|
||||
|
||||
<h2>An interesting coincidence</h2>
|
||||
|
||||
<p>
|
||||
The angle $\beta$ constructed by this method coincides <em>exactly</em>
|
||||
with that of <a href="trisect-jamison.html">Lindberg's construction</a>,
|
||||
where $\beta$ is given as:
|
||||
\[
|
||||
\beta
|
||||
= \frac{1}{4} \alpha + \arctan
|
||||
\frac{\sin\frac{\alpha}{4}}{2+\cos\frac{\alpha}{4}}.
|
||||
\]
|
||||
One way to verify that the seemingly different expressions
|
||||
for $\beta$ are in fact identical,
|
||||
is to compare their derivatives. In both cases we have:
|
||||
\[
|
||||
\frac{d\beta}{d\alpha} =
|
||||
\frac{3(1 + \cos\frac{\alpha}{4})}{2(5 + 4\cos\frac{\alpha}{4})}.
|
||||
\]
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on June 10, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,152 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>
|
||||
William R. Raiford, <i>An approximate trisection</i>,
|
||||
American Mathematical Monthly,
|
||||
vol. 68, no. 9, Nov 1961, p. 917.
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="450" height="400">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;210,365">
|
||||
<param name="e[2]" value="A;point;fixed;410,365">
|
||||
<param name="e[3]" value="pt0;point;fixed;410,0;none;none">
|
||||
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|
||||
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|
||||
<param name="e[6]" value="B;point;circleSlider;cir1,0,300;red;red">
|
||||
<param name="e[7]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[8]" value="OB;line;connect;O,B;none;none;blue">
|
||||
<param name="e[9]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
|
||||
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|
||||
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|
||||
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|
||||
<param name="e[13]" value="li1;line;connect;B,C;none;none;lightGray">
|
||||
<param name="e[14]" value="li2;line;extend;B,C,B,C;none;none;lightGray">
|
||||
<param name="e[15]" value="T;point;intersection;li0,li2">
|
||||
<param name="e[16]" value="OT;line;connect;O,T;none;none;red">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[17]" value="p1;point;fixed;240,385;none;none">
|
||||
<param name="e[18]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[19]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[20]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[21]" value="l2;line;chord;O,T,c1;none;none;none">
|
||||
<param name="e[22]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[23]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[24]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[25]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[26]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line $OT$ is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
The construction described in the article cited at the top of the page,
|
||||
is quite straightforward. Consider the angle $AOB$ represented by the
|
||||
circular arc $AB$ centered at $O$, as shown in the diagram above.
|
||||
To trisect $AOB$ do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Erect a perpendicular to $OA$ at $A$ (shown in green).
|
||||
|
||||
<li>
|
||||
Construct the bisector $OC$ of the angle $AOB$.
|
||||
|
||||
<li>
|
||||
Connect $B$ to $C$ and extend to intersect the green line at a point $T$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$,
|
||||
respectively. One may verify that
|
||||
\[
|
||||
\beta
|
||||
= \arctan \Big( \sin\alpha - (1 - \cos\alpha)
|
||||
\cot \big( \frac{3}{4}\alpha \big) \Big)
|
||||
= \frac{1}{3}\alpha + \frac{1}{2^3\cdot3^4} \alpha^3 + O(\alpha^5)
|
||||
= \frac{1}{3}\alpha + \frac{1}{648} \alpha^3 + O(\alpha^5).
|
||||
\]
|
||||
|
||||
<p>
|
||||
The error
|
||||
$
|
||||
\ds e(\alpha) = \beta - \frac{\alpha}{3}
|
||||
$
|
||||
is monotonically increasing in $\alpha$.
|
||||
The worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
$e(\pi/2)$ = 0.0063 radians = 0.361 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.06 radians = 3.435 degrees.
|
||||
|
||||
<p>
|
||||
<span class="name">Raiford</span>, whose affiliation is given as IBM,
|
||||
states that he has calculated
|
||||
the error in increments of one degree in an IBM 709. Computers
|
||||
were novelties when that article was published.
|
||||
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a>
|
||||
on June 14, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -0,0 +1,399 @@
|
|||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
<html>
|
||||
<head>
|
||||
<!-- fix buggy IE8, especially for mathjax -->
|
||||
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||||
<title>An angle trisection</title>
|
||||
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
|
||||
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
|
||||
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
|
||||
</script>
|
||||
|
||||
</head>
|
||||
<body style="visibility:hidden">
|
||||
|
||||
<h1>An angle trisection</h1>
|
||||
|
||||
<h4>Construction by
|
||||
<a href="mailto:mhs210@hotmail.com">Mark Stark</a>
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="600" height="600">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<!-- The angle AOB -->
|
||||
<param name="e[1]" value="O;point;fixed;300,300">
|
||||
<param name="e[2]" value="A;point;fixed;585,300">
|
||||
<param name="e[3]" value="C1;circle;radius;O,A;none;none;lightGray;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;C1,70,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
|
||||
<!-- The inner circle -->
|
||||
<param name="e[7]" value="A';point;fixed;395,300">
|
||||
<param name="e[8]" value="C2;circle;radius;O,A';none;none;lightGray;none">
|
||||
<param name="e[9]" value="L1;line;chord;OB,C2;none;none;none">
|
||||
<param name="e[10]" value="B';point;first;L1">
|
||||
<param name="e[11]" value="A'B';line;connect;A',B';none;none;green">
|
||||
|
||||
<!-- Points E and D -->
|
||||
<param name="e[12]" value="E;point;circleSlider;C2,600,150;red;red">
|
||||
<param name="e[13]" value="C3;circle;radius;A',E;none;none;lightGray;none">
|
||||
<param name="e[14]" value="L2;line;chord;A'B',C3;none;none;none">
|
||||
<param name="e[15]" value="D;point;first;L2">
|
||||
|
||||
<param name="e[16]" value="L3;line;chord;D,E,C1;none;none;none">
|
||||
<param name="e[17]" value="G;point;last;L3">
|
||||
<param name="e[18]" value="EG;line;connect;E,G;none;none;lightGray">
|
||||
<param name="e[19]" value="L4;line;chord;G,O,C1;none;none;none">
|
||||
<param name="e[20]" value="T;point;last;L4">
|
||||
<param name="e[21]" value="GT;line;connect;G,T;none;none;red">
|
||||
|
||||
<param name="e[22]" value="p1;point;fixed;325,300;none;none">
|
||||
<param name="e[23]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[24]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[25]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[26]" value="l2;line;chord;O,T,c1;none;none;none">
|
||||
<param name="e[27]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[28]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<param name="e[29]" value="p2;point;fixed;325,300;none;none">
|
||||
<param name="e[30]" value="c2;circle;radius;O,p2;none;none;none;none">
|
||||
<param name="e[31]" value="l3;line;chord;O,T,c2;none;none;none">
|
||||
<param name="e[32]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[33]" value="l4;line;chord;OB,c2;none;none;none">
|
||||
<param name="e[34]" value="q4;point;first;l4;none;none">
|
||||
<param name="e[35]" value="s2;sector;sector;O,q3,q4;none;none;black;yellow">
|
||||
|
||||
<param name="pivot" value="O">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
Drag the point $B$ to change the angle $AOB$
|
||||
(but stay on the upper half of the semicircle).<br>
|
||||
Drag the point “$E$” to change the radius of the circle
|
||||
centered at $A'$. Note how little $G$ is affected by the choice of $E$.<br>
|
||||
Press “r” to reset the diagram to its initial state.<br>
|
||||
The red line ($O$T) is an approximate trisector of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
This construction, due to Mark Stark, was announced
|
||||
<a href="http://mathforum.org/kb/message.jspa?messageID=1088614">
|
||||
in an message</a> in the <code>geometry.puzzles</code> newsgroup on
|
||||
Jun 20, 2002.
|
||||
Scroll to the bottom of that page to view the related discussion thread.
|
||||
|
||||
<p>
|
||||
The construction is unusual
|
||||
because one of the steps involves an arbitrary choice. It is interesting
|
||||
that the result is quite insensitive to that choice.
|
||||
|
||||
<p>
|
||||
Here I have paraphrased Mark's construction but
|
||||
differences from the original are cosmetic. The error analysis is mine.
|
||||
|
||||
<p>
|
||||
Consider the circular arc $AB$ on the circle $C$ centered at $O$,
|
||||
shown in the diagram above.
|
||||
Assume the angle $AOB$ is between 0 and 180 degrees.
|
||||
To trisect $AOB$, do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Draw a circle $C'$ centered at $O$ with a radius 1/3 $OA$.
|
||||
Mark $A'$ and $B'$ its intersections with the line segments $OA$ and $OB$,
|
||||
respectively.
|
||||
|
||||
<li>
|
||||
Draw the line $A'B'$ (shown in green).
|
||||
|
||||
<li>
|
||||
Draw a circle $C''$ centered at $A'$ with an arbitrary(!) radius.
|
||||
The accuracy of the trisection will be affected by the choice
|
||||
of the radius, albeit only slightly. Best results are obtained
|
||||
when the angle $EA'D$ (see the next step) is close to one third
|
||||
of the angle $AOB$.
|
||||
|
||||
<li>
|
||||
Let $D$ be the intersection of $C''$ with the line segment $A'B'$.<br>
|
||||
Let $E$ be the intersection of $C''$ with the $C'$, as shown.
|
||||
|
||||
<li>
|
||||
Draw the line $ED$ and extend to the intersection point $G$ with the the
|
||||
circle $C$.
|
||||
|
||||
<li>
|
||||
Draw the diameter $GOT$.
|
||||
|
||||
</ol>
|
||||
|
||||
The line $OT$ is an approximate trisector of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
In the diagram below, I have duplicated the previous diagram
|
||||
and added the lines $OE$ and $EA'$ which are not needed in
|
||||
the construction, but are needed for the error analysis.
|
||||
|
||||
<p>
|
||||
You may zoom and translate the diagram to examine its details.
|
||||
To zoom in, grab the point $B'$ with the mouse
|
||||
and move it away from $O$. To translate,
|
||||
grab $O$ and move it around. As always, type “r” to
|
||||
reset the diagram to its initial state.
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="600" height="600">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<!-- The angle AOB -->
|
||||
<param name="e[1]" value="O;point;fixed;300,300">
|
||||
<param name="e[2]" value="A;point;fixed;585,300">
|
||||
<param name="e[3]" value="C1;circle;radius;O,A;none;none;lightGray;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;C1,70,0;red;red">
|
||||
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
|
||||
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
|
||||
|
||||
<!-- The inner circle -->
|
||||
<param name="e[7]" value="A';point;fixed;395,300">
|
||||
<param name="e[8]" value="C2;circle;radius;O,A';none;none;lightGray;none">
|
||||
<param name="e[9]" value="L1;line;chord;OB,C2;none;none;none">
|
||||
<param name="e[10]" value="B';point;first;L1">
|
||||
<param name="e[11]" value="A'B';line;connect;A',B';none;none;green">
|
||||
|
||||
<!-- Points E and D -->
|
||||
<param name="e[12]" value="E;point;circleSlider;C2,600,150;red;red">
|
||||
<param name="e[13]" value="C3;circle;radius;A',E;none;none;lightGray;none">
|
||||
<param name="e[14]" value="L2;line;chord;A'B',C3;none;none;none">
|
||||
<param name="e[15]" value="D;point;first;L2">
|
||||
|
||||
<param name="e[16]" value="L3;line;chord;D,E,C1;none;none;none">
|
||||
<param name="e[17]" value="G;point;last;L3">
|
||||
<param name="e[18]" value="EG;line;connect;E,G;none;none;lightGray">
|
||||
<param name="e[19]" value="L4;line;chord;G,O,C1;none;none;none">
|
||||
<param name="e[20]" value="T;point;last;L4">
|
||||
<param name="e[21]" value="GT;line;connect;G,T;none;none;red">
|
||||
|
||||
<param name="e[22]" value="p1;point;fixed;325,300;none;none">
|
||||
<param name="e[23]" value="c1;circle;radius;O,p1;none;none;none;none">
|
||||
<param name="e[24]" value="l1;line;chord;OA,c1;none;none;none">
|
||||
<param name="e[25]" value="q1;point;first;l1;none;none">
|
||||
<param name="e[26]" value="l2;line;chord;O,T,c1;none;none;none">
|
||||
<param name="e[27]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[28]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
|
||||
<param name="e[29]" value="p2;point;fixed;325,300;none;none">
|
||||
<param name="e[30]" value="c2;circle;radius;O,p2;none;none;none;none">
|
||||
<param name="e[31]" value="l3;line;chord;O,T,c2;none;none;none">
|
||||
<param name="e[32]" value="q3;point;first;l3;none;none">
|
||||
<param name="e[33]" value="l4;line;chord;OB,c2;none;none;none">
|
||||
<param name="e[34]" value="q4;point;first;l4;none;none">
|
||||
<param name="e[35]" value="s2;sector;sector;O,q3,q4;none;none;black;yellow">
|
||||
|
||||
<param name="e[36]" value="EA;;line;connect;E,A';none;none;lightGray">
|
||||
<param name="e[37]" value="OE;;line;connect;O,E;none;none;lightGray">
|
||||
|
||||
<param name="pivot" value="O">
|
||||
|
||||
</applet>
|
||||
</td></tr>
|
||||
<tr><td>
|
||||
<b>
|
||||
To zoom in, grab the point $B'$ with the mouse
|
||||
and move it away from $O$.<br>
|
||||
To translate, grab $O$ and move it around.<br>
|
||||
Type “r” to return to the initial state.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOT$,
|
||||
respectively. We will show that $\beta \approx \frac{1}{3}\alpha$.
|
||||
|
||||
<p>
|
||||
The construction leaves the size of the circle $C''$ (centered at $A'$)
|
||||
unspecified. We parametrize the circle by the position of the point
|
||||
$E$ along the arc $A'B'$, or more precisely, by the value $\gamma$ of the
|
||||
angle angle $B'A'E$.
|
||||
Thus $\gamma=0$ when $E$ coincides with $B'$
|
||||
and $\gamma=\alpha/2$ (easy to verify) when E coincides with $A'$.
|
||||
|
||||
<p>
|
||||
Since the angles $B'A'E$ and $B'OE$ subtend the arc $B'E$ of the
|
||||
circle $C'$, then the angle $B'OE$ is $2\gamma$. Therefore the angle
|
||||
$EOA'$ is $\alpha - 2\gamma$. But $EOA'$ is the vertex angle of the
|
||||
triangle $EOA'$, therefore the base angle $OEA'$ is $\frac{1}{2}(\pi - \alpha +
|
||||
2\gamma)$.
|
||||
|
||||
<p>
|
||||
In the isosceles triangle $DA'E$, the vertex angle is $\gamma$, therefore
|
||||
the base angle $DEA'$ is $\frac{1}{2}(\pi - \gamma)$.
|
||||
|
||||
<p>
|
||||
Putting the assertions of the two previous paragraphs together, we
|
||||
calculate the angle $OED$:
|
||||
\[
|
||||
OED = OEA' - DEA' = \frac{1}{2}(3\gamma - \alpha)
|
||||
\]
|
||||
|
||||
<p>
|
||||
In the triangle $GOE$, we have just computed the angle at $E$ (because
|
||||
$OED$ is the same as $OEG$). Let us write $x$ for the angle at $G$.
|
||||
Then $x$ may be computed by applying the law of sines and noting
|
||||
that the ratio of the sides $OG$ to $OE$ is 3. We get:
|
||||
$3\sin x = \sin\frac{1}{2}(3\gamma-\alpha)$.
|
||||
|
||||
<p>
|
||||
The external angle $EOT$ of the triangle $GOE$ equals the sum of the
|
||||
remaining internal angles, that is:
|
||||
\[
|
||||
EOT = x + \frac{1}{2}(3\gamma-\alpha).
|
||||
\]
|
||||
|
||||
On the other hand,
|
||||
\[
|
||||
EOT
|
||||
= EOB' - TOB'
|
||||
= EOB' - (A'OB' - A'OT)
|
||||
= 2\gamma - (\alpha - \beta).
|
||||
\]
|
||||
|
||||
We see then $x + \frac{1}{2}(3\gamma-\alpha) = 2\gamma - (\alpha - \beta)$,
|
||||
whence $x = \beta + \gamma/2 - \beta/2$. This leads to the equation:
|
||||
\[
|
||||
3\sin\big(\beta + \frac{1}{2}\gamma - \frac{1}{2}\beta\big)
|
||||
=
|
||||
\sin\frac{1}{2}(3\gamma-\alpha),
|
||||
\]
|
||||
which we may solve for $\beta$:
|
||||
\[
|
||||
\beta =
|
||||
\frac{1}{3}\alpha + \frac{1}{6}(\alpha-3\gamma)
|
||||
-
|
||||
\arcsin\bigg[
|
||||
\frac{1}{3}\sin\Big(\frac{\alpha-3\gamma}{2} \Big)
|
||||
\bigg]
|
||||
\]
|
||||
|
||||
<p>
|
||||
As expected, the constructed angle,
|
||||
$\beta$, depends on the original angle $\alpha$ we well as the choice
|
||||
of $\gamma$. Let us express this dependence as $\beta = \tau(\alpha,\gamma)$.
|
||||
Expanding $\tau$ in power series we get:
|
||||
\[
|
||||
\beta = \tau(\alpha,\gamma)
|
||||
= \frac{1}{3}\alpha
|
||||
+ \frac{4}{3}\Big(\frac{\alpha-3\gamma}{6} \Big)^3
|
||||
- \frac{4}{5}\Big(\frac{\alpha-3\gamma}{6} \Big)^7
|
||||
+ O\bigg( \Big(\frac{\alpha-3\gamma}{6}\Big)^9 \bigg).
|
||||
\]
|
||||
|
||||
The term with exponent 5 is absent in the series expansion; that's not a typo.
|
||||
|
||||
<h3>On the choice of $\gamma$</h3>
|
||||
|
||||
<p>
|
||||
We see that $\tau(\alpha,\alpha/3) = \alpha/3$, that is,
|
||||
the construction produces an <em>exact trisection</em>
|
||||
with the choice $\gamma=\alpha/3$.
|
||||
Of course, constructing such a $\gamma$
|
||||
is equivalent to solving the original trisection problem, therefore
|
||||
that is not an option. On the other hand, a constructible $\gamma$ that
|
||||
comes close to $\alpha/3$ will serve just fine. The function $\tau$ is
|
||||
not very sensitive to the variations of $\gamma$ as is evident from:
|
||||
\[
|
||||
\frac{\partial \tau(\alpha,\gamma)}{\partial \gamma}
|
||||
=
|
||||
\frac{1}{2} \bigg(
|
||||
\frac{3\cos 3x}{\sqrt{9 - \sin^2 3x}} -1 \bigg),
|
||||
\]
|
||||
where I have let $x=(\alpha-3\gamma)/6$ to simplify the notation.
|
||||
As noted above, best
|
||||
results are achieved when $\gamma$ is close to $\alpha/3$. Even
|
||||
with a not-so-optimal choice of $\gamma=\alpha/4$ we get $x=\alpha/24$.
|
||||
With such a choice, the value of partial derivative in the range
|
||||
$0 \le \alpha \le \pi/2$ does not exceed 0.01, indicating that
|
||||
the value of the function is essentially independent of $\gamma$
|
||||
on that range.
|
||||
|
||||
<p>
|
||||
<b>An excellent choice</b> for $\gamma$ is obtained as follows.
|
||||
In Step 3 of the construction, first select the point $D$ on
|
||||
the line segment $A'B'$ such that $A'D = \frac{1}{3} A'B'$.
|
||||
Then draw the circle $C''$ with center $A'$ passing through $D$.
|
||||
One may verify that this results in an angle $\gamma$ given by:
|
||||
\[
|
||||
\hat{\gamma} = \frac{1}{2} \alpha
|
||||
- \arcsin\Big( \frac{1}{3} \sin\frac{\alpha}{2} \Big)
|
||||
= \frac{1}{3} \alpha + \frac{1}{2\cdot3^4} \alpha^3 + O(\alpha^7).
|
||||
\]
|
||||
|
||||
Then, the constructed angle is:
|
||||
\[
|
||||
\beta = \tau(\alpha,\hat{\gamma})
|
||||
= \tau\bigg(\alpha, \frac{1}{2} \alpha
|
||||
- \arcsin\Big( \frac{1}{3} \sin\frac{\alpha}{2} \Big)
|
||||
\bigg)
|
||||
= \frac{1}{3} \alpha - \frac{1}{2^4\cdot3^{13}} \alpha^9 +
|
||||
O(\alpha^{13}).
|
||||
\]
|
||||
The construction error, $e(\alpha) = \frac{1}{3}\alpha-\beta$,
|
||||
is monotone increasing. Since $e(\alpha) = O(\alpha^9)$,
|
||||
we expect it to be very small.
|
||||
Indeed, the worst error on the interval $0 \le \alpha \le \pi/2$ is
|
||||
the incredibly small
|
||||
$e(\pi/2)$ = 0.00000226 radians = 0.00013 degrees.
|
||||
The worst error on the interval $0 \le \alpha \le \pi$ is
|
||||
$e(\pi)$ = 0.00103 radians = 0.0592 degrees.
|
||||
|
||||
<p>
|
||||
Despite its extraordinary accuracy, this is <em>not</em>
|
||||
among my favorite trisection methods because the points $D$ and $E$ are
|
||||
too close to each other for locating the point $G$ reliably. For practical
|
||||
purposes, should there be such a need, I would much rather use a more
|
||||
robust, albeit less accurate, method.
|
||||
|
||||
<hr width="60%">
|
||||
<p>
|
||||
<em>This applet was created by
|
||||
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
|
||||
using
|
||||
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
|
||||
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
|
||||
Applet</a> on
|
||||
July 26, 2002.
|
||||
<br>The error analysis was thoroughly revised and extensive
|
||||
cosmetic changes were made on June 7, 2010.
|
||||
</em>
|
||||
<p>
|
||||
|
||||
<table width="100%">
|
||||
<tr>
|
||||
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
|
||||
<td align="right" style="width:200px;">
|
||||
<a href="http://validator.w3.org/check?uri=referer">
|
||||
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
|
||||
<a href="http://jigsaw.w3.org/css-validator/check/referer">
|
||||
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
|
||||
</td></tr>
|
||||
</table>
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -303,6 +303,7 @@ function jToG(
|
|||
cmdr
|
||||
|
||||
function invisible(cname: string): boolean
|
||||
if adapParams.config?.showall then return false
|
||||
cname is '0' or cname is 'none'
|
||||
|
||||
function joyce2rgb(cname: string, backgroundRGB?: RGB): RGB
|
||||
|
|
@ -378,16 +379,6 @@ classHandler: Record<JoyceClass, ClassHandler> :=
|
|||
aux := name + 'aUx'
|
||||
parts[0].push name
|
||||
switch method
|
||||
/free|fixed/
|
||||
commands.push `${name} = (${args.scalar?.join ','})`
|
||||
if method is 'fixed'
|
||||
callbacks.push (api: AppletObject) => api.setFixed name, true
|
||||
'perpendicular'
|
||||
// Note only the two-point option implemented so far
|
||||
unless args.subpoints return
|
||||
[center, direction] := args.subpoints
|
||||
// Note clockwise 90° rotation (3π/2) confirmed in Joyce source
|
||||
commands.push `${name} = Rotate(${direction}, 3*pi/2, ${center})`
|
||||
'angleDivider'
|
||||
// Note doesn't yet handle plane argument
|
||||
unless args.subpoints return
|
||||
|
|
@ -406,6 +397,27 @@ classHandler: Record<JoyceClass, ClassHandler> :=
|
|||
`${aux}4 = Rotate(${start}, ${aux}3/${n}, ${center})`
|
||||
`${name} = Intersect(${destination}, Ray(${center}, ${aux}4))`
|
||||
auxiliaries.push ...[2..4].map (i) => `${aux}${i}`
|
||||
'extend'
|
||||
unless args.subpoints then return
|
||||
sp := args.subpoints
|
||||
direction .= `UnitVector(Vector(${sp[0]},${sp[1]}))`
|
||||
if args.line and (
|
||||
not args.point or args.point[0] !== args.subpoints[0])
|
||||
direction = `UnitVector(${args.line[0]})`
|
||||
displacement := `Distance(${sp[2]}, ${sp[3]})*${direction}`
|
||||
commands.push `${name} = Translate(${sp[1]}, ${displacement})`
|
||||
'first'
|
||||
unless args.subpoints then return
|
||||
commands.push `${name} = ${args.subpoints[0]}`
|
||||
/fixed|free/
|
||||
commands.push `${name} = (${args.scalar?.join ','})`
|
||||
if method is 'fixed'
|
||||
callbacks.push (api: AppletObject) => api.setFixed name, true
|
||||
'foot'
|
||||
pt := args.subpoints
|
||||
unless pt then return
|
||||
commands.push
|
||||
`${name} = ClosestPoint(Line(${pt[1]},${pt[2]}), ${pt[0]})`
|
||||
'intersection'
|
||||
// Checking Joyce source, means intersection of lines, not
|
||||
// intersection of line segments
|
||||
|
|
@ -413,6 +425,9 @@ classHandler: Record<JoyceClass, ClassHandler> :=
|
|||
l1 := `Line(${args.subpoints[0]},${args.subpoints[1]})`
|
||||
l2 := `Line(${args.subpoints[2]},${args.subpoints[3]})`
|
||||
commands.push `${name} = Intersect(${l1},${l2})`
|
||||
'last'
|
||||
unless args.subpoints then return
|
||||
commands.push `${name} = ${args.subpoints.at(-1)}`
|
||||
'lineSegmentSlider'
|
||||
segment .= args.line?[0]
|
||||
unless segment
|
||||
|
|
@ -424,35 +439,28 @@ classHandler: Record<JoyceClass, ClassHandler> :=
|
|||
if args.scalar and args.scalar.length
|
||||
callbacks.push (api: AppletObject) =>
|
||||
api.setCoords name, ...args.scalar as XYZ
|
||||
'first'
|
||||
unless args.subpoints then return
|
||||
commands.push `${name} = ${args.subpoints[0]}`
|
||||
'last'
|
||||
unless args.subpoints then return
|
||||
commands.push `${name} = ${args.subpoints.at(-1)}`
|
||||
'extend'
|
||||
unless args.subpoints then return
|
||||
sp := args.subpoints
|
||||
direction .= `UnitVector(Vector(${sp[0]},${sp[1]}))`
|
||||
if args.line and (
|
||||
not args.point or args.point[0] !== args.subpoints[0])
|
||||
direction = `UnitVector(${args.line[0]})`
|
||||
displacement := `Distance(${sp[2]}, ${sp[3]})*${direction}`
|
||||
commands.push `${name} = Translate(${sp[1]}, ${displacement})`
|
||||
'vertex'
|
||||
commands.push
|
||||
`${name} = Vertex(${args.polygon?[0]},${args.scalar?[0]})`
|
||||
'midpoint'
|
||||
if args.line
|
||||
commands.push `${name} = Midpoint(${args.line[0]})`
|
||||
else
|
||||
commands.push
|
||||
`${name} = Midpoint(${args.point?[0]},${args.point?[1]})`
|
||||
'foot'
|
||||
'perpendicular'
|
||||
// Note only the two-point option implemented so far
|
||||
unless args.subpoints return
|
||||
[center, direction] := args.subpoints
|
||||
// Note clockwise 90° rotation (3π/2) confirmed in Joyce source
|
||||
commands.push `${name} = Rotate(${direction}, 3*pi/2, ${center})`
|
||||
'proportion'
|
||||
pt := args.subpoints
|
||||
unless pt then return
|
||||
len := `Distance(${pt[2]},${pt[3]})*Distance(${pt[4]},${pt[5]})`
|
||||
+ `/ Distance(${pt[0]},${pt[1]})`
|
||||
direction := `UnitVector(Vector(${pt[6]}, ${pt[7]}))`
|
||||
commands.push `${name} = Translate(${pt[6]}, ${len}*${direction})`
|
||||
'vertex'
|
||||
commands.push
|
||||
`${name} = ClosestPoint(Line(${pt[1]},${pt[2]}), ${pt[0]})`
|
||||
`${name} = Vertex(${args.polygon?[0]},${args.scalar?[0]})`
|
||||
|
||||
line: (name, method, args) =>
|
||||
return := freshCommander()
|
||||
|
|
|
|||
|
|
@ -1,4 +1,4 @@
|
|||
export const flags = ['commands', 'color'] as const
|
||||
export const flags = ['color', 'commands', 'showall'] as const
|
||||
export type FlagType = (typeof flags)[number]
|
||||
export type ConfigType = Partial<Record<FlagType, boolean>>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,11 +1,8 @@
|
|||
import {flags} from ./adapptypes.ts
|
||||
|
||||
console.log('arrived')
|
||||
boxes := ['commands', 'color']
|
||||
cache := await browser.storage.local.get boxes
|
||||
console.log('Found', cache)
|
||||
cache := await browser.storage.local.get flags
|
||||
|
||||
for each box of boxes
|
||||
for each box of flags
|
||||
checkbox := document.getElementById(box) as HTMLInputElement
|
||||
unless checkbox then continue
|
||||
checkbox.checked = cache[box] ?? false
|
||||
|
|
|
|||
Loading…
Reference in New Issue