APMO 2020 P1

Discussion on Asian Pacific Mathematical Olympiad (APMO)
Soumitro_Shovon
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Joined:Fri Aug 21, 2020 11:39 am
APMO 2020 P1

Unread post by Soumitro_Shovon » Thu Dec 03, 2020 9:19 pm

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

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Ahnaf_Akif
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Joined:Fri Dec 04, 2020 6:44 pm

Re: APMO 2020 P1

Unread post by Ahnaf_Akif » Tue Dec 08, 2020 2:22 pm

By angle chasing,
$\angle EAC=\angle B= \angle EAD$
So $ACED$ is cyclic and by radical axis theorem on $(ACBF),(EDBF),(ACED)$ we are done. :)

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