I have no idea how to solve this problem, but I'm pretty sure that it could be made easier by using Roots of Unity! Help would be appreciated!
Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find $PA_1^2 + PA_2^2 + \dots + PA_{11}^2$.
You can take the $A_j$ to have coordinates $2\zeta$ where $\zeta$ runs through the eleventh roots of unity. Then $P$ will have coordinates $3u$ for some complex $u$ with $|u|=1$. So $$|PA_j|^2=|3u-2\zeta|^2 =(3u-2\zeta)(3\overline u-2\overline\zeta)=13-6(\zeta\overline u+\overline \zeta u).$$ It should be easy to add these up over all zeta.