IN-TST 2010 Problem 7

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$ . Suppose $AD$ and $BC$ meet in $F$ . Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma$ is the circumcircle of triangle $EGH$ , prove that $FE$ is tangent to $\Gamma$ .