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Diagonal Product (Posted on 2005-10-20) Difficulty: 4 of 5
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)

See The Solution Submitted by owl    
Rating: 4.3333 (3 votes)

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Solution Solution Comment 3 of 3 |
 
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. 
 

  Posted by Bractals on 2005-10-20 17:35:27
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