INMO 1999 Problem 4

Let $\Gamma$ and $\Gamma'$ be two concentric circles. Let $ABC$ and $A'B'C'$ be any two equilateral triangles inscribed in $\Gamma$ and $\Gamma'$ respectively. If $P$ and $P'$ are any two points on $\Gamma$ and $\Gamma'$ respectively, show that $$P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2.$$