IT-CESENATICO 2015 Problem 5

Let $AB$ be a chord of a circle $\Gamma$ and let $C$ be a point on the segment $AB$ . Let $r$ be a line through $C$ which intersects $\Gamma$ at the points $D,E$ ; suppose that $D,E$ lie on different sides with respect to the perpendicular bisector of $AB$ . Let $\Gamma_D$ be the circumference which is externally tangent to $\Gamma$ at $D$ and touches the line $AB$ at $F$ . Let $\Gamma_E$ be the circumference which is externally tangent to $\Gamma$ at $E$ and touches the line $AB$ at $G$ . Prove that $CA=CB$ if and only if $CF=CG$ .