OMM 2018 Problem 6

Let $ABC$ be an acute-angled triangle with circumference $\Omega$ . Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$ . Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$ . Prove that the center of the circle passing through $I$ , $M'$ , $N'$ lies on the altitude of triangle $ABC$ from $A$ .

Proposed by Victor Domínguez and Ariel García