OMCC 2009 Problem 2

\item Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$ . Consider a circle $\Gamma$ contained in $\Gamma_1$ and $\Gamma_2$ , which is tangent to both of them at $D$ and $E$ respectively. Let $C$ be one of the intersection points of line $AB$ with $\Gamma$ , $F$ be the intersection of line $EC$ with $\Gamma_2$ and $G$ be the intersection of line $DC$ with $\Gamma_1$ . Let $H$ and $I$ be the intersection points of line $ED$ with $\Gamma_1$ and $\Gamma_2$ respectively. Prove that $F$ , $G$ , $H$ and $I$ are on the same circle.