Question 79
The inscribed circle of an acute triangle $ABC$ (with incenter $I$) touches sides $AB$ and $BC$ in points $C_1$ and $A_1$ respectively. Let $M$ be the midpoint of the side $AC$, and $N$ be the midpoint of an arc $ABC$ of the circumcircle of $ABC$. Also, let $P$ be the projection of a point $M$ to the segment $A_1C_1$. Prove the points $I$, $P$, and $N$ lie on the same line.
(Anton Trygub)
**Fig. 35**
(Anton Trygub)
**Fig. 35**