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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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Some Thoughts Possible solution | Comment 2 of 3 |

By inspection from smaller polygons,

4 sides: 2S+D , where S is a side, and D is the diameter

6 sides 2S+2S1+D, where S1 is the line of intermediate length

8 sides 2S+2S1+2S2+D, etc.

Since all the vertices are on the circumference, these can all be seen as right triangles, with D as the hypotenuse, so their measure is:

2^2+2*(2*sin(12))^2+2*(2*sin(24))^2+2*(2*sin(36))^2+2*(2*sin(48))^2+2*(2*sin(60))^2+2*(2*sin(72))^2+2*(2*sin(84))^2 = 34, plus

2*(2*sin(6))^2+2*(2*sin(18))^2+2*(2*sin(30))^2+2*(2*sin(42))^2+2*(2*sin(54))^2+2*(2*sin(66))^2+2*(2*sin(78))^2 =26

for a total of 60.

  Posted by broll on 2018-04-17 13:36:26
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