 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Diagonal Product (Posted on 2005-10-20) 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) Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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