Circumscribed quadrilateral and equal segments

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Let $O$ and $P$ are the incenter and intersection points of the diagonals of the circumscribed quadrilateral $ABCD$ respectively. Though $P$ is drawn line $l$, perpendicular to $OP$. $M$ and $N$ are the intersection points of $l$ with the sides of $ABCD$. Prove that $P$ is the middle of $MN$.

(Оффтоп)

This problem is too easy. Only inscribed quadrilaterals and Brianchon theorem solves it.