INMO 2011 Problem 5

Let $A B C D$ be a quadrilateral inscribed in a circle $\Gamma$. Let $E, F, G, H$ be the midpoints of the arcs $A B, B C, C D, D A$ of the circle $\Gamma$. Suppose $A C \cdot B D=E G \cdot F H$. Prove that $A C, B D$, $E G, F H$ are concurrent.