CGMO 2003 Problem 3

As shown in the figure, quadrilateral $ABCD$ is inscribed in a circle with $AC$ as its diameter, $BD \perp AC,$ and $E$ the intersection of $AC$ and $BD.$ Extend line segment $DA$ and $BA$ through $A$ to $F$ and $G$ respectively, such that $DG \parallel{} BF.$ Extend $GF$ to $H$ such that $CH \perp GH.$ Prove that points $B, E, F$ and $H$ lie on one circle.