Question 95
Let $ABC$ be a non-isosceles triangle with incenter $I$, circumcenter $O$ and a point $D$ on segment $BC$ such that $(BID)$ cuts segment $AB$ at $E \neq B$ and $(CID)$ cuts segment $AC$ at $F \neq C$. Circle $(DEF)$ cuts segments $AB$, $AC$ again at $M$, $N$. Let $P = IB \cap DE$ and $Q = IC \cap DF$. Prove that $EN$, $FM$, $PQ$ are parallel and the median of vertex $I$ in triangle $IPQ$ bisects the arc $BAC$ of $(O)$.