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<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">
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<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">
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<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">
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<param name="e[31]" value="l2;line;chord;OT,c1;none;none;none">
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<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>
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