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<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>
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