KöMaL 2012 B4444

The points $E$ and $G$, respectively, are obtained by rotating vertex $D$ of a square ABCD about $A$ and vertex $B$ (lying opposite $D$) about $C$ outwards, through the same acute angle $\alpha$. The intersection of lines $AE$ and $BG$is $F$. Line $GD$ intersects the circles GCB and BDE again at points $L$and $M$, respectively. Prove that the points $A$, $B$, $F$, $L$, $M$ are cyclic.