CGMO 2015 Problem 6

Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$ . $F$ is a point on $\Gamma_1$ , the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$ . $MG$ is a line tangent to $\Gamma_2$ at $G$ . Prove that $MF=MG$ .