APMO 2018 Problem 1

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samiul_samin
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APMO 2018 Problem 1

Unread post by samiul_samin » Thu Jan 10, 2019 11:03 pm

Let $H$ be the ortho center of the triangle $ABC$.Let $M$ and $N$ be the midpoint of the sides $AB$ and $AC$ ,respectively.Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.

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