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 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)

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 No proof, but ... (spoiler) | Comment 1 of 3

Each line segment is a chord of the circumscribing circle, subtending 360*i/n degrees, where i goes from 1 to n-1. The length of such a chord is twice the sine of half the subtended arc. So, using Prod to signify what is usually denoted by a capital Pi, the formula would be Prod{i=1 to n-1} 2 sin(180 i / n deg). Tabulated for n = 3 to 25, this comes out to

`3       3.000000000000000000000000000000000000000000000000000000000000014       4.000000000000000000000000000000000000000000000000000000000000075       5.000000000000000000000000000000000000000000000000000000000000086       6.00000000000000000000000000000000000000000000000000000000000027       7.000000000000000000000000000000000000000000000000000000000000638       8.000000000000000000000000000000000000000000000000000000000000649       9.0000000000000000000000000000000000000000000000000000000000008410      10.000000000000000000000000000000000000000000000000000000000000811      11.0000000000000000000000000000000000000000000000000000000000015812      12.0000000000000000000000000000000000000000000000000000000000022413      13.0000000000000000000000000000000000000000000000000000000000027514      14.0000000000000000000000000000000000000000000000000000000000030515      15.0000000000000000000000000000000000000000000000000000000000036116      16.000000000000000000000000000000000000000000000000000000000004817      17.0000000000000000000000000000000000000000000000000000000000047618      18.0000000000000000000000000000000000000000000000000000000000046719      19.0000000000000000000000000000000000000000000000000000000000057820      20.0000000000000000000000000000000000000000000000000000000000071821      21.000000000000000000000000000000000000000000000000000000000010222      22.0000000000000000000000000000000000000000000000000000000000091223      23.0000000000000000000000000000000000000000000000000000000000092924      24.0000000000000000000000000000000000000000000000000000000000108625      25.00000000000000000000000000000000000000000000000000000000001551`

indicating that the non-zeros near the ends of these numbers are the result of rounding errors, and that the simplified formula for the product is actually N.

For N=3, for example, the sides of the triangle are sqrt(3). Only two of the sides are multiplied together, so the result is 3.  For a square (N=4), the sides are sqrt(2) and the diagonal is 2. The product of two sides and the diagonal is 4.

4   point 13
5   Pi=atan(1)*4:Dr=Pi/180
10   for N=3 to 25
20     P=1
30     for I=1 to N-1
40      P=P*2*sin(180*I*Dr/N)
50     next I
60     print N,P
70   next N

Edited on October 20, 2005, 2:09 pm
 Posted by Charlie on 2005-10-20 13:54:01

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