Glen Whitney
f1696120dc
This change implements several additional construction methods, including the first polygon ones. In particular, it now allows arbitrary strings as entity names, even ones that are not allowed as GeoGebra identifiers, using captions to show the original entity names. In addition, line arguments are interpreted as a pair of point arguments as needed. Resolves #6. Resolves #30. Resolves #31.
136 lines
5.1 KiB
HTML
136 lines
5.1 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 equilateral triangle inscribed in a rectangle</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>An equilateral triangle inscribed in a rectangle</h1>
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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="410" height="370">
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<param name="background" value="ffffff">
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<param name="title" value="An equilateral triangle inscribed in a rectangle">
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<!-- the moving mechanism -->
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<param name="e[1]" value="O;point;fixed;290,320">
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<param name="e[2]" value="U1;point;fixed;510,320">
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<param name="e[3]" value="V1;point;perpendicular;O,U1">
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<param name="e[4]" value="U;point;angleDivider;U1,O,V1,3">
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<param name="e[5]" value="V;point;angleDivider;V1,O,U1,3">
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<param name="e[6]" value="circ1;circle;radius;O,U">
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<param name="e[7]" value="li1;line;parallel;U,O,U1">
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<param name="e[8]" value="li2;line;parallel;V,O,V1">
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<param name="e[9]" value="W;point;intersection;li1,li2">
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<param name="e[10]" value="VW;line;connect;V,W;0;0;lightGray">
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<param name="e[11]" value="@;point;lineSegmentSlider;V,W,0,220;red;red">
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<param name="e[12]" value="li3;line;parallel;@,O,U1">
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<param name="e[13]" value="li4;line;chord;circ1,li3">
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<param name="e[14]" value="X1;point;first;li4">
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<!-- the triangle -->
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<param name="e[15]" value="A;point;fixed;50,320">
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<param name="e[16]" value="V2;point;perpendicular;A,U1">
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<param name="e[17]" value="li5;line;parallel;A,O,X1">
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<param name="e[18]" value="X2;point;last;li5">
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<param name="e[19]" value="X;point;extend;A,X2,A,X2">
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<param name="e[20]" value="tri1;polygon;equilateralTriangle;X,A">
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<param name="e[21]" value="Y;point;vertex;tri1,3">
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<param name="e[22]" value="B;point;midpoint;X,Y">
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<param name="e[23]" value="ABC;polygon;equilateralTriangle;A,B">
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<param name="e[24]" value="C;point;vertex;ABC,3">
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<!-- the rectangle -->
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<param name="e[25]" value="D;point;foot;B,A,U1">
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<param name="e[26]" value="F;point;foot;C,A,V2">
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<param name="e[27]" value="FE;line;parallel;F,A,D">
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<param name="e[28]" value="E;point;last;FE">
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<param name="e[29]" value="rect;polygon;quadrilateral;A,D,E,F;0;0;black;0">
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<param name="e[30]" value="ADB;polygon;triangle;A,D,B;0;0;0;pink">
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<param name="e[31]" value="ACF;polygon;triangle;A,C,F;0;0;0;pink">
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<param name="e[32]" value="BCE;polygon;triangle;B,C,E;0;0;0;cyan">
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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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Slide the “@” up and down to change the geometry.<br>
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Press “r” to reset the diagram to its initial state.<br>
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Proposition: The blue area equals the sum of the two pink areas.
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</b>
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</td></tr></table>
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<h2>Problem statement</h2>
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<p>
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The diagram above shows an equilateral triangle inscribed in a rectangle
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in such a way that the two have a vertex in common. This subdivides the
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rectangle into four disjoint triangles.
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The original equilateral triangle is shown in white
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in the diagram; the other three are shown in color.
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<p>
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<b>Proposition</b>
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<em>
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The area of the blue triangle equals the sum
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of the areas of the two pink triangles.
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</em>
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<p>
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The trigonometric proof is quite straightforward. I don't
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know of a classical proof <i>a la</i> <span class="name">Euclid</span>.
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(Well, actually I haven't tried much.)
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If you can think of a neat non-trigonometric proof, let me know. I will
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put it here with due credit.
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<p>
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This problem appeared as a conjecture
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<a href="http://mathforum.org/kb/thread.jspa?forumID=129&messageID=1083967">in an article</a>
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in the <code>geometry.puzzles</code> newsgroup on March 15, 1997.
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<p>
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<b>Note added January 8, 2017:</b>
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Here is a
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<a href="inscribed-equilateral-solution.html">clever solution</a>
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that <b>Peter Renz</b> sent me a in December 2016. Thanks, Peter!
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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>
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on July 2, 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">Geometry Problems and Puzzles</a></td>
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<td align="right" style="width:200px;">
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<a href="http://validator.w3.org/check?uri=referer">
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<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
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<a href="http://jigsaw.w3.org/css-validator/check/referer">
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<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
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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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