KöMaL 2007 A431

Triangle ABC is not isosceles. The incenter is $I$, the excenter is $O$. The incircle touches the sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Lines $FD$ and $AC$intersect at $P$, lines $DE$ and $AB$ intersect at $Q$. The midpoints of segments $EP$ amd $FQ$ are $M$ and $N$, respectively. Prove that $MN$ and $OI$ are perpendicular.