Question

Let $A_1$, $A_2$, $\dots$, $A_{12}$ be twelve equally spaced points on a circle with radius 1. Find
\[(A_1 A_2)^2 + (A_1 A_3)^2 + \dots + (A_{11} A_{12})^2.\]
(The sum includes the square of the distance between any pair of points, so the sum includes $\binom{12}{2} = 66$ terms.)