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 \(1 \le i < j \le 12.)\)
We can use the sine law to find P1 P2
By the sine law, P1 P2 = sin 15. We can also find P1 P_3 = sin 60, P1 P4 = sin 90, etc.
So (P1 P2)^2 + (P1 P3)^2 + … + (P11 P12)^2 = 12 (sin 15)^2 + 12 (sin 30)^2 + 12 (sin 45)^2 + … + 12 (sin 90)^2 = 42.
