RO-TST 2004 Problem 11

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$ , the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$ . Prove that the lines $IP$ and $MN$ are perpendicular.

Alternative formulation. The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$ , $CA$ and $AB$ in $A^{\prime}$ , $B^{\prime}$ and $C^{\prime}$ , respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$ , the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$ , and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$ . Prove that the lines $IP$ and $MN$ are perpendicular.