trisection of angle

Given an angle of measure (http://planetmath.org/AngleMeasure) $\alpha$ such that , one can construct an angle of measure $\frac{\alpha }{3}$ using a compass and a ruler (http://planetmath.org/MarkedRuler) with one mark on it as follows:

1. Construct a circle $c$ with the vertex (http://planetmath.org/Vertex5) $O$ of the angle as its center. Label the intersections of this circle with the rays of the angle as $A$ and $B$. Mark the length $OB$ on the ruler.

2. Draw the ray $\overrightarrow{AO}$.

3. Use the marked ruler to determine $C\in c$ and $D\in \overrightarrow{AO}$ such that $CD=OB$ and $B$, $C$, and $D$ are collinear. Draw the line segment $\overline{BD}$. Then the angle measure of $\mathrm{\angle }CDO$ 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.)

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

• Since $\mathrm{\angle }AOB$ is an exterior angle of $\mathrm{△}BOD$, we have that $m(\mathrm{\angle }AOB)=m(\mathrm{\angle }OBD)+m(\mathrm{\angle }ODB)$;

• Since $OC=OB=CD$, we have that $\mathrm{△}BOC$ and $\mathrm{△}OCD$ are isosceles triangles;

• Since the angles of an isosceles triangle are congruent, $m(\mathrm{\angle }OBC)=m(\mathrm{\angle }OCB)$ and $m(\mathrm{\angle }COD)=m(\mathrm{\angle }CDO)$;

• Since $\mathrm{\angle }OCB$ is an exterior angle of $\mathrm{△}OCD$, we have that $m(\mathrm{\angle }OCB)=m(\mathrm{\angle }COD)+m(\mathrm{\angle }CDO)$;

• Note that $\mathrm{\angle }OBC=\mathrm{\angle }OBD$ and $\mathrm{\angle }ODB=\mathrm{\angle }CDO$;

• Thus,

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 :

• If , then use the construction given above.

• If , then trisect an angle of measure $\beta -\frac{\pi }{2}$ and add on an angle of measure $\frac{\pi }{6}$ to the result.

• If , then trisect an angle of measure $\beta -\pi$ and add on an angle of measure $\frac{\pi }{3}$ to the result.

• If , 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.