Circles $T_1$ and $T_2$ intersect at two points $B $ and $C$, and $BC$ is the diameter of $T_1$. Construct a tangent line to circle $T_1$ at $C$ intersecting $T_2$ at another point $A$. Line $AB$ meets $T_1$ again at $E $and line $CE $ meets $T_2$ again at $F $. Let $H $ be an arbitrary point on segment $AF $. Line $HE $meets $T_2$ again at $G $, and $BG $ meets $AC $ at $D $.

Prove that $$AH / HF=AC / CD $$

## CGMO 2002/4

- Ananya Promi
**Posts:**36**Joined:**Sun Jan 10, 2016 4:07 pm**Location:**Naogaon, Bangladesh

- Raiyan Jamil
**Posts:**138**Joined:**Fri Mar 29, 2013 3:49 pm

### Re: CGMO 2002/4

**Correction:**$G$ lies on $T_1$, not $T_2$.

**Solution:**$I$ be a point on line $AF$ such that it is the reflection of $D$ under $AB$. We get $HEBI$ is cyclic. So, $AH \times AI=AE \times AB=AF^2$ from which we can get the result.

**A smile is the best way to get through a tough situation, even if it's a fake smile.**

- Ananya Promi
**Posts:**36**Joined:**Sun Jan 10, 2016 4:07 pm**Location:**Naogaon, Bangladesh

### Re: CGMO 2002/4

I found this in EGMO(Euclidean geometry in mathematical Olympiad). Then it's their mistake. Not mine.