CGMO 2002 Problem 4

Circles $O_1$ and $O_2$ interest at two points $B$ and $C,$ and $BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A.$ We join $AB$ to intersect circle $O_1$ at point $E,$ then join $CE$ and extend it to intersect circle $O_2$ at point $F.$ Assume $H$ is an arbitrary point on line segment $AF.$ We join $HE$ and extend it to intersect circle $O_1$ at point $G,$ and then join $BG$ and extend it to intersect the extend line of $AC$ at point $D.$ Prove that $$\frac{AH}{HF} = \frac{AC}{CD}.$$