Trisecting the Angle/Archimedean Spiral

Theorem

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of an Archimedean spiral.

In the figure, the blue line is an Archimedean spiral.

Let $\angle AOB$ be the angle to be trisected, where $OB$ is on the polar axis.

Trisect the line segment $OA$ such that $OC = \dfrac 1 3 OA$.

Draw a circle with $O$ as the center and $OC$ as the radius.

Let $D$ be the intersection of the circle and the spiral.

Then:

$\angle DOB = \dfrac 1 3 \angle AOB$

Let the equation of the Archimedean spiral be $r = a \theta$.

Then:

$\ds \angle DOB$

$=$

$\ds \frac {OD} a$

$\ds $

$=$

$\ds \frac {OA/3} a$

$\ds $

$=$

$\ds \frac 1 3 \angle AOB$

$\blacksquare$

Use of the Archimedean spiral to trisect an angle was a standard technique for mathematicians following Archimedes.