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

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