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

ro = rcos(180k/n)

proof: Consider the following figure.

O is the center of the circle. Pk is a point on an {n/k} star which is inscribed in the circle. The chords from Pk are the two lines of the {n/k} star which go through Pk. Po 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 Pk to the center of the circle bisects the angle at Pk. It is also well known that the angle at Po 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).

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