PlanetMath: trisection of angle

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trisection of angle

(Algorithm)

Given an angle of measure $\alpha$ such that $0<\alpha \le \frac{\pi}{2}$ , one can construct an angle of measure $\frac{\alpha}{3}$ using a compass and a ruler with one mark on it as follows:

Construct a circle with the vertex of the angle as its center. Label the intersections of this circle with the rays of the angle as and . Mark the length on the ruler.

$\begin{pspicture}(-2,-3)(3,3) \rput[l](-2,0){.} \rput[r](3,2){.} \psline{->}(0,0... ...t[a](2.1,-0.3){$A$} \rput[a](1.7,1.4){$B$} \rput[a](0,-2.2){$c$} \end{pspicture}$
Draw the ray $\overrightarrow{AO}$ .

$\begin{pspicture}(-5,-3)(3,3) \rput[l](-5,0){.} \rput[r](3,2){.} \psline{->}(0,0... ...t[a](2.1,-0.3){$A$} \rput[a](1.7,1.4){$B$} \rput[a](0,-2.2){$c$} \end{pspicture}$
Use the marked ruler to determine $C\in c$ and $D\in \overrightarrow{AO}$ such that and , , and are collinear. Draw the line segment $\overline{BD}$ . Then the angle measure of $\angle CDO$ is $\frac{\alpha}{3}$ . (The line segment $\overline{OC}$ is drawn in red. Having this line segment drawn is useful for reference purposes for the justification of the construction.)

$\begin{pspicture}(-5,-3)(3,3) \rput[l](-5,0){.} \rput[r](3,2){.} \psline{<->}(-5... ...0,-2.2){$c$} \rput[r](-2,0.6){$C$} \rput[a](-3.923445,-0.3){$D$} \end{pspicture}$

Let denote the measure of an angle. Then this construction is justified by the following:

Since $\angle AOB$ is an exterior angle of $\triangle BOD$ , we have that $m(\angle AOB)=m(\angle OBD)+m(\angle ODB)$ ;
Since , we have that $\triangle BOC$ and $\triangle OCD$ are isosceles triangles;
Since the angles of an isosceles triangle are congruent, $m(\angle OBC)=m(\angle OCB)$ and $m(\angle COD)=m(\angle CDO)$ ;
Since $\angle OCB$ is an exterior angle of $\triangle OCD$ , we have that $m(\angle OCB)=m(\angle COD)+m(\angle CDO)$ ;
Note that $\angle OBC=\angle OBD$ and $\angle ODB=\angle CDO$ ;
Thus,
$\begin{displaymath}\begin{array}{rl} \alpha & =m(\angle AOB) \ & =m(\angle OBD... ...+m(\angle CDO)+m(\angle CDO) \ & =3m(\angle CDO). \end{array}\end{displaymath}$

Note that, since angles of measure $\frac{\pi}{6}$ , $\frac{\pi}{3}$ , and $\frac{\pi}{2}$ are constructible using compass and straightedge, this procedure can be extended to trisect any angle of measure $\beta$ such that $0<\beta\le 2\pi$ :

If $0<\beta\le\frac{\pi}{2}$ , then use the construction given above.
If $\frac{\pi}{2}<\beta\le\pi$ , then trisect an angle of measure $\beta-\frac{\pi}{2}$ and add on an angle of measure $\frac{\pi}{6}$ to the result.
If $\pi<\beta\le\frac{3\pi}{2}$ , then trisect an angle of measure $\beta-\pi$ and add on an angle of measure $\frac{\pi}{3}$ to the result.
If $\frac{3\pi}{2}<\beta\le 2\pi$ , then trisect an angle of measure $\beta-\frac{3\pi}{2}$ and add on an angle of measure $\frac{\pi}{2}$ to the result.

This construction is attributed to Archimedes.

Bibliography

1: Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.

"trisection of angle" is owned by Wkbj79.

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Cross-references: straightedge, constructible, congruent, angles of an isosceles triangle, isosceles triangles, exterior angle, reference, angle measure, line segment, collinear, marked ruler, length, rays, intersections, center, circle, compass, angle
There are 3 references to this entry.

This is version 8 of trisection of angle, born on 2007-06-18, modified 2007-06-23.
Object id is 9616, canonical name is TrisectionOfAngle.
Accessed 1469 times total.

Classification:

AMS MSC:	51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
	01A20 (History and biography :: History of mathematics and mathematicians :: Greek, Roman)

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