MEMO 2015 I3

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$ , respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$ , respectively. Prove that points $C$ , $D$ , $F$ , and $G$ lie on a circle.