Suppose we have the N vertices of a regular N-gon inscribed in a circle of radius 1. Select one vertex W and draw line segments from W to each of the other N-1 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
(z-x_1)(z-x_2)(z-x_3) ... (z-x_n) = z^n - 1
Since x_n = 1,
(z-x_1)(z-x_2)(z-x_3) ... (z-x_[n-1])
z^n - 1 z^n - 1
= --------- = ---------
z - x_n z - 1
= z^(n-1) + z^(n-2) + ... + z + 1
Let z = 1 be our vertex W and we have
(1-x_1)(1-x_2)(1-x_3) ... (1-x_[n-1]) = n
or
|1-x_1||1-x_2||1-x_3| ... |1-x_[n-1]| = n
Therefore, the product of the n-1 chord lengths is n.
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Posted by Bractals
on 2005-10-20 17:35:27 |
Solution
