## CGMO 2002/4

For discussing Olympiad level Geometry Problems
Ananya Promi
Posts: 36
Joined: Sun Jan 10, 2016 4:07 pm
Location: Naogaon, Bangladesh

### CGMO 2002/4

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 $$AH / HF=AC / CD$$

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.
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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.