OMM 2014 Problem 4

Problem 4

Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$ . Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$ , $F$ on $CD$ such that $E$ is midpoint of $DF$ , and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$ ). Prove that the circumference through $D$ , $F$ and $G$ is tangent to $BG$ .