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# Help with Geometry!!!

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A regular dodecagon \$P_1 P_2 P_3 \dotsb P_{12}\$ is inscribed in a circle with radius 1.  Compute

\[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\]
(The sum includes all terms of the form \$(P_i P_j)^2,\$ where \$1 \le i < j \le 12.\$ )

Please, I want hints, not a solution!

Oct 3, 2020

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There are 12 digaonals that have a length of P_1 P_2, which from the Sine Law, is sin (15 degrees).  There are 12 diagonals that have a legnth of P_1 P_3, which from the Sine Law, is sin (30 degrees).  We can appy the same reasoning to the other diagonals, which gives us a total sum of

(12 sin 15)^2 + (12 sin 30)^2 + (12 sin 45)^2 + (12 sin 60)^2 + (12 sin 75)^2 + (12 sin 90)^2 + (12 sin 105)^2 + (12 sin 120)^2 + (12 sin 135)^2 + (12 sin 150)^2 + (12 sin 175)^2 = 864.

Since we have double-counted, the answer is 864/2 = 432.

Oct 3, 2020
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+149
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Pangolin14  Oct 4, 2020
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I think all those squared distances are relatively easy to find using trigonometry.  Cosine Rule

Is that a good enough hint?

Oct 5, 2020