Question 112
Circles $O_1$ and $O_2$ intersect 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 $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume that $H$ is an arbitrary point on the line segment $AF$. We join $HE$ and extend it to intersect circle $O_1$ at point $G$, and join $BG$ and extend it to intersect the extended line of $AC$ at point $D$.
Prove that $\frac{AH}{HF} = \frac{AC}{CD}$.
Prove that $\frac{AH}{HF} = \frac{AC}{CD}$.