**Theorem 4.2**: (Coxeter
[4]) The perpendicular distance
from the center of the circle to the lines which make up the star is

**proof**: Consider the following figure.

O is the center of the circle. P_{k} is a point on an
{n/k} star which is inscribed in the circle. The chords from
P_{k} are the two lines of the {n/k} star which go through
P_{k}. P_{o} is the foot of the line from the center
of the circle perpendicular to the chord. It is easy to show that the
radius from P_{k} to the center of the circle bisects the
angle at P_{k}. It is also well known that the angle at
P_{o} is a right angle, so it follows that the angle at O is
180k/n, and the distance from O to Po is rcos(180k/n).