Question 5
Let a scalene triangle $ABC$ have circumcircle $(O)$ and incircle $(I)$. Let $E=BI\cap AC$ and $F=CI\cap AB$. The circle through $E$ tangent to the line $OB$ at $B$ meets $(O)$ again at $M$; the circle through $F$ tangent to $OC$ at $C$ meets $(O)$ again at $N$. Let $P,Q$ be the second intersections of $ME,NF$, respectively, with $(O)$. Put $K=EF\cap BC$, $G=PQ\cap BC$, and $H=PQ\cap EF$. Prove that the median from $G$ in triangle $GHK$ is perpendicular to $OI$.