## APMO 2018 Problem 1

Discussion on Asian Pacific Mathematical Olympiad (APMO)
samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### APMO 2018 Problem 1

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 $F J = F A$.

thczarif
Posts: 17
Joined: Mon Sep 25, 2017 11:27 pm
Let $P$ be a point on the common tangent to $\left(BMH\right)$ and $\left(CMH\right)$ such that $P$ is further from $BC$ than $H$. Then $\angle{MHP}=\angle{MKH}=\angle{MBH}=\frac{\pi}{2}-A$and similarly $\angle{NHP}=\frac{\pi}{2}-A$, so $\angle{MHN}=\pi-2A$. Then $\angle{MJN}=\frac{\pi}{2}+\frac{\pi-2A}{2}=\pi-A$, so $AMJN$ is cyclic. Now observe that since $MN$ is parallel to $KL$, $\angle{FMN}=\angle{FKL}=\angle{MKH}=\frac{\pi}{2}-A$and similarly $\angle{FNM}=\frac{\pi}{2}-A$, but these angle relations are satisfied for $F$ inside $\triangle{AMN}$ only when $F$ is the circumcenter of $\triangle{AMN}$, so $F$ is the circumcenter of cyclic quadrilateral $AMJN$. It follows that $FJ=FA$.