Question 125
Let $\Gamma$ be a circle, $AB$ a chord of it, $C$ a point inside the segment $AB$, and $r$ a line through $C$ such that, denoting by $D$ and $E$ the intersections of $r$ with $\Gamma$, these lie on opposite sides of the perpendicular bisector of $AB$. Let $\Gamma_{D}$ be the circle externally tangent to $\Gamma$ at $D$ and tangent to $AB$ at a point $F$, and $\Gamma_{E}$ the circle externally tangent to $\Gamma$ at $E$ and tangent to $AB$ at a point $G$. Prove that $CA = CB$ if and only if $CF = CG$.