KöMaL 2005 A383

The diameter $AB$ of the circle $k$ is perpendicular to the line $\ell$, which does not pass through either $A$ or $B$. Let $C$ be a point of $\ell$ outside $k$, and let $D$ denote the other intersection of the line $AC$ with the circle. Draw one of the tangents from $C$ to $k$and denote the point of contact by $E$. Let $F$ be the intersection of the lines $BE$ and $\ell$, and finally let $G$ denote the intersection of the line segment $AF$ with $k$. Show that the line $DF$ passes through the reflection of the point $G$ about $AB$.