Suppose we have the N vertices of a regular Ngon inscribed in a circle of radius 1. Select one vertex W and draw line segments from W to each of the other N1 vertices. What is the total product of the lengths of these line segments?
(old problem  original author unknown)
By the product formula given, the formula would products of sines from zero to pi/2 (or 180 degrees). I have the idea that an inductive technique could be used, as sin(pi/2  x) = cos (x). Then two sines can be combined into one, as sin (2x) = 2 sin(x) cos(x), or sin(2x) = 2 sin(x)sin(pi/2  x), thereby reducing the number of intervals or the value of N, to half the original. This would work for values of N that are powers of 2 (using induction multiplicatively rather than additively), but I can't see how to formalize this, or where then N comes from in the formula. (I see where the halving of the intervals offsets the powers of two.)

Posted by Charlie
on 20051020 14:21:02 