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The Devil 30-gon (Posted on 2018-04-17) Difficulty: 3 of 5
If A1, A2, A3, . . . , A30 are the vertices of a regular 30-gon inscribed in a unit circle, then find

|A1A2|2 + |A1A3|2 + |A1A4|2 + ... + |A1A30|2

No Solution Yet Submitted by Danish Ahmed Khan    
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solution Comment 3 of 3 |
My trig is pretty sketchy, pun intended, but here goes.

Using the law of cosines and assuming a unit circle with 30-gon side s, s^2 = 2-2cos(x) where x is angle formed by the two radii/legs.

Each of the 30 distance squares can be paired with one 180 degrees away.  Since cos(x) = -cos(x+180), all the cosine sums cancel and we're left with 30*2 = 60 as the desired answer. 

  Posted by xdog on 2018-04-17 19:03:54
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