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CGMO 2002/4

Posted: Sat Sep 30, 2017 12:02 pm
by Ananya Promi
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 $$AH / HF=AC / CD $$

Re: CGMO 2002/4

Posted: Sun Oct 01, 2017 2:59 pm
by Raiyan Jamil
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.

Re: CGMO 2002/4

Posted: Wed Oct 04, 2017 10:49 pm
by Ananya Promi
I found this in EGMO(Euclidean geometry in mathematical Olympiad). Then it's their mistake. Not mine.