Problem

Source: 2017 VMO Problem 3

Tags: geometry, circumcircle



Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. $AH$ intersects $(O)$ at $D$ ($D\ne A$). a) Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN\perp OH$. b) The lines $DE$, $DF$ intersect $(O)$ at $P,Q$ respectively ($P\ne D,Q\ne D$). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ($R\ne A, S\ne A$). Prove that $BP,CQ,RS$ are concurrent.