CWMO 2010 Problem 6

$\Delta ABC$ is a right-angled triangle, $\angle C = 90^{\circ}$ . Draw a circle centered at $B$ with radius $BC$ . Let $D$ be a point on the side $AC$ , and $DE$ is tangent to the circle at $E$ . The line through $C$ perpendicular to $AB$ meets line $BE$ at $F$ . Line $AF$ meets $DE$ at point $G$ . The line through $A$ parallel to $BG$ meets $DE$ at $H$ . Prove that $GE = GH$ .