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)
Let the n vertices represent the n roots of unity
x_k = e^(i*[2*k*PI/n]) for k = 1,2,...,n
Then
(zx_1)(zx_2)(zx_3) ... (zx_n) = z^n  1
Since x_n = 1,
(zx_1)(zx_2)(zx_3) ... (zx_[n1])
z^n  1 z^n  1
=  = 
z  x_n z  1
= z^(n1) + z^(n2) + ... + z + 1
Let z = 1 be our vertex W and we have
(1x_1)(1x_2)(1x_3) ... (1x_[n1]) = n
or
1x_11x_21x_3 ... 1x_[n1] = n
Therefore, the product of the n1 chord lengths is n.

Posted by Bractals
on 20051020 17:35:27 