China Girls' Mathematical Olympiad

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. (posed by Liu Jiangfeng)