VN-TST 2019 Problem 5

Given a scalene triangle $ABC$ inscribed in the circle $(O)$ . Let $(I)$ be its incircle and $BI,CI$ cut $AC,AB$ at $E,F$ respectively. A circle passes through $E$ and touches $OB$ at $B$ cuts $(O)$ again at $M$ . Similarly, a circle passes through $F$ and touches $OC$ at $C$ cuts $(O)$ again at $N$ . $ME,NF$ cut $(O)$ again at $P,Q$ . Let $K$ be the intersection of $EF$ and $BC$ and let $PQ$ cuts $BC$ and $EF$ at $G,H$ , respectively. Show that the median correspond to $G$ of the triangle $GHK$ is perpendicular to $IO$ .