269 lines
10 KiB
HTML
269 lines
10 KiB
HTML
<!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>An angle trisection</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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</head>
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<body style="visibility:hidden">
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<h1>An angle trisection</h1>
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<h4>Construction by
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<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a></h4>
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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="570" height="360">
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<param name="background" value="ffffff">
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<param name="title" value="An angle trisection">
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<!-- the angle AOB -->
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<param name="e[1]" value="O;point;fixed;260,310">
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<param name="e[2]" value="A;point;fixed;520,310">
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<param name="e[3]" value="OA;line;connect;O,A;none;none;blue">
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<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
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<param name="e[5]" value="B;point;circleSlider;C,20,0;red;red">
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<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
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<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
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<!-- point P quadrisects the arc AB -->
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<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
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<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
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<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
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<param name="e[11]" value="P;point;first;Ot2">
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<param name="e[12]" value="AP;line;connect;A,P;none;none;green">
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<!-- the arc A'OB' -->
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<param name="e[13]" value="A';point;fixed;455,310">
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<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
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<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
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<param name="e[16]" value="B';point;first;t9">
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<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
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<!-- P' -->
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<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
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<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
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<param name="e[20]" value="P';point;first;t4">
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<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
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<param name="e[22]" value="s2;point;first;s1;none;none">
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<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
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<param name="e[24]" value="A'P';line;connect;A',P';none;none;green">
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<!-- angle marker -->
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<param name="e[25]" value="p1;point;fixed;290,310;none;none">
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<param name="e[26]" value="c1;circle;radius;O,p1;none;none;none;none">
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<param name="e[27]" value="l1;line;chord;OA,c1;none;none;none">
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<param name="e[28]" value="q1;point;first;l1;none;none">
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<param name="e[29]" value="l2;line;chord;s3,c1;none;none;none">
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<param name="e[30]" value="q2;point;first;l2;none;none">
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<param name="e[31]" value="S1;sector;sector;O,q1,q2;none;none;black;orange">
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<param name="e[32]" value="l3;line;chord;OB,c1;none;none;none">
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<param name="e[33]" value="q3;point;first;l3;none;none">
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<param name="e[34]" value="S2;sector;sector;O,q2,q3;none;none;black;yellow">
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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 the point $B$ to change the angle $AOB$.<br>
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Press “r” to reset the diagram to its initial state.<br>
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The red line is an approximate trisector of the angle $AOB$.
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</b>
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</td></tr></table>
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<H2>The Basic Construction</H2>
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<p>
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Here is a very simple straightedge and compass
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construction of an approximate angle trisector due to
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<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a>.
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<p>
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Let us represent the angle by
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the circular arc $AB$ centered at $O$; see the diagram above.
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The angle's size may be anything from 0 to 180 degrees.
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To trisect, do:
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<ol>
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<li>
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Quadrisect the angle $AOB$, that is, divide it into four
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equal parts. The arc $AP$ in the diagram above represents one quarter
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of the original arc $AB$. Let $L$ be the length of the chord $AP$ (shown in green).
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<li>
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Draw a circular arc (shown in orange)
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centered at $O$ and radius 3/4 of $OA$. Mark $A'$ and $B'$ its intersections
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with the rays $OA$ and $OB$, respectively.
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<li>
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Swing an arc (not shown) of radius $L$ centered at $A'$ and mark $P'$ its intersection with
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the arc $A'B'$, as shown.
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</ol>
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<p>
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The line $OP'$ is an approximate trisector of the angle $AOB$.
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<h2>Error Analysis</h2>
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<p>
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Let $\alpha$ and $\beta=\tau(\alpha)$ be the sizes of the angles $AOB$ and $A'OP'$,
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respectively. It is straightforward to show that
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\[
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\beta
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= 2 \arcsin\big(\frac{4}{3}\sin\frac{\alpha}{8}\big)
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= \frac{\alpha}{3} + \frac{7}{2^7\cdot3^4}\alpha^3 + O(\alpha^5)
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= \frac{\alpha}{3} + \frac{7}{10368}\alpha^3 + O(\alpha^5).
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\]
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<!-- The first two terms of the series are the same as those
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in trisect-dunham.html. The third terms are different.
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b_baker := 2*arcsin(4/3*sin(a/8));
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series(b_baker,a);
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b_durham := a/2 - arctan(sin(a/4 - arcsin(sin(a/4)/2))*4/3);
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series(b_durham, a);
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-->
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<p>
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The error
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$
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\ds e(\alpha) = \tau(\alpha) - \frac{\alpha}{3}
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$
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is monotonically increasing in $\alpha$.
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The worst error on the interval $0 \le \alpha \le \pi/2$ is
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$e(\pi/2)$ = 0.002695 radians = 0.154 degrees.
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The worst error on the interval $0 \le \alpha \le \pi$ is
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$e(\pi)$ = 0.0237 radians = 1.360 degrees.
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<h2>Iterative Improvement</h2>
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<p>
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As we see in the asymptotic expansion shown above, the
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angle $\tau(\alpha)$ is slightly larger than the target value of $\alpha/3$.
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Making three copies of the constructed angle, and putting them
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end-to-end as in arcs $A'P'$, $P'P''$, and $P''P'''$ shown in the diagram below,
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we arrive at the endpoint $P'''$ which is very slightly off the point $B'$,
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and just outside the arc $A'B'$. The constructible angle $B'OP'''$ is exactly
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three times the error $e(\alpha)$.
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If we were able to trisect $B'OP'''$ exactly, then we
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would know the error, and consequently will have achieved
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the exact trisection of the original angle.
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Of course the exact trisection of $B'OP'''$ is impossible in general, but we
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may repeat the method outlined in the <em>Basic Construction</em> above
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to obtain an <em>approximate</em> trisection of $B'OP'''$,
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which will yield $ \tau\big(3\tau(\alpha) - \alpha\big) $,
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and consequently an improved trisection $\tau_{\mathrm{improved}}(\alpha)$
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of the original angle:
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\[
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\tau_{\mathrm{improved}}(\alpha) = \tau(\alpha) - \tau\big(3\tau(\alpha) - \alpha\big)
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= \frac{\alpha}{3} - \frac{7^4}{2^{28}\cdot3^{13}} \alpha^9 +
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O(\alpha^{11}).
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\]
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The error
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$ \ds e_{\mathrm{improved}}(\alpha) = \frac{\alpha}{3} - \tau_{\mathrm{improved}}(\alpha)$
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is monotonically increasing in $\alpha$. In particular,
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$e_{\mathrm{improved}}(\pi/2) = 1.5\times 10^{-9}$ radians
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$ = 8.6\times10^{-8}$ degrees.
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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="570" height="360">
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<param name=background value="ffffff">
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<param name=title value="An angle trisection">
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<!-- the angle AOB -->
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<param name="e[1]" value="O;point;fixed;260,310">
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<param name="e[2]" value="A;point;fixed;520,310">
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<param name="e[3]" value="OA;line;connect;O,A">
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<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
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<param name="e[5]" value="B;point;circleSlider;C,50,0">
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<param name="e[6]" value="OB;line;connect;O,B">
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<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
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<!-- point P quadrisects the arc AB -->
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<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
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<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
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<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
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<param name="e[11]" value="P;point;first;Ot2">
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<param name="e[12]" value="OP;line;connect;O,P;none;none;none">
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<!-- the arc A'OB' -->
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<param name="e[13]" value="A';point;fixed;455,310">
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<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
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<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
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<param name="e[16]" value="B';point;first;t9">
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<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
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<!-- P' -->
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<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
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<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
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<param name="e[20]" value="P';point;first;t4">
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<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
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<param name="e[22]" value="s2;point;first;s1;none;none">
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<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
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<!-- P'' -->
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<param name="e[24]" value="t5;circle;radius;P',A,P;none;none;none;none">
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<param name="e[25]" value="t6;line;bichord;t5,C';none;none;none">
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<param name="e[26]" value="P'';point;first;t6">
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<!-- P'''
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Note the trailing spaces after P'''. These become a part of the label!
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-->
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<param name="e[27]" value="t7;circle;radius;P'',A,P;none;none;none;none">
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<param name="e[28]" value="t8;line;bichord;t7,C';none;none;none">
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<param name="e[29]" value="P''' ;point;first;t8">
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<param name="e[30]" value="OP''';line;connect;O,P''' ;none;none;black">
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<param name="e[31]" value="u1;sector;sector;O,B',P''' ;none;none;none;magenta">
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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 the point $B$ to change the angle $AOB$.<br>
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Press “r” to reset the diagram to its initial state.<br>
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The red line is an approximate trisector of the angle $AOB$.<br>
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The arcs $P'P''$ and $P''P'''$ are copies of $A'P'$. The endpoint $P'''$
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is just slightly off the point $B'$.<br>
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The (very small and nearly indiscernible)
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angle $B'OP'''$ is three times the trisection error.
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</b>
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</td></tr></table>
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<hr width="60%">
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<p>
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<em>This applet was created by
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<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
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using
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<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
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<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
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Applet</a> on
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May 31, 2010.
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</em>
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<p>
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<table width="100%">
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<tr>
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<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
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<td align="right" style="width:200px;">
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</td></tr>
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</table>
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</body>
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</html>
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