Theorem 4.4b: In an {n/k} star inscribed in a circle of radius r, if 1 < j < n/2 then the distance from the Pj points to the Pi points on a line of the star is
If the points are on the same side of Po use the minus sign. If they are on opposite sides, use the plus sign.
proof 2: If Pj and Pi are on the same side of Po then we have the following figure
The Pj are points on the {n/j} star, and we have drawn in the circle through the Pj points. We are assuming that i < j, so the Pi point is inside this circle. We have drawn the two lines through the {n/j} star which meet at Pi, and M is the midpoint of the line joining these two Pj points.
By Theorem 4.3, the distance from O to Pj is rj = rcos(180k/n)sec(180j/n). The distance from Pj to M is then rj times the sine of the angle from M through O to Pj. To find that angle, note that the angle from O through Pj to Pi is half of a Pj angle or 90 - 180j/n. The angle from M through Pi to Pj is half of a Pi angle or 90 - 180i/n, so the angle from Pj through Pi to O is the supplement or 90 + 180i/n. Thus the angle from M through O to is 180 - (90 - 180j/n) - (90 + 180i/n) = 180j/n - 180i/n.
If Pj and Pi are on the opposite sides of Po then we have the following figure
Again, the angle from Pj through O to Pi is 180 - (90 - 180j/n) - (90 - 180i/n) = 180j/n + 180i/n. Hence, in this case the angle from M through O to Pj is 180 - (180j/n + 180i/n). However, sin(180 - (180j/n + 180i/n)) = sin(180j/n + 180i/n), and we have that the distance from M to Pj is
depending on whether and Pi are on the opposite sides of Po or not.
In either case, since the angle from M through to Pi is the complement of half of a Pi angle, the distance from Pj to Pi is