Theorem 4.3: (Zeitler [14]) Let an {n/k} star be placed in a coordinatized plane in such a way that the center of the star is at the origin and the polar coordinates of one of the points is (r, 90). Let rj be the distance from the points Pj to the center of the circle. Then

rj = rcos(180k/n)sec(180j/n)

The angle at which the Pj points are found is

90 + 360j/n

if j is congruent to k mod 2


90 + 180(2j - 1)/n

if j is not congruent to k mod 2

i = 1, . . . , n, j = 1, . . . , k

proof: The formulas for the angles follow from the fact that the angles for the points on the {n/k-1} star are halfway between the angles for the {n/k} stars and iteration.

If Pj is a point on the {n/j} star on the line in the {n/k} star we have the following figure.

Since Pk is a point on an {n/k} star, and O is the center of the circle, it follows that the angle at O is 180j/n, and the result follows.

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