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 \(1 \le i < j \le 12\).)
+23251 If C is the center of the circle, then angle(P1CP2) = 360o / 12 = 30o.
To find the length of chord P1P2, we can use Heron's formula: c2 = a2 + b2 - 2·a·b·cos(C)
---> c2 = 12 + 12 - 2·1·1·cos(30o)
= 1 + 1 - 2·sqrt(3)/2
= 2 - sqrt(3)
Since all chords equal each other, the sum is: 12( 2 - sqrt(3) ) = 24 - 12·sqrt(3)
