Question 124
Let $ABC$ be an acute triangle with incenter $I$. The line through $I$ parallel to $AC$ intersects $AB$ at $M$; and the line through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $ABC$ at $X$ and $Y$. Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of triangle $XYZ$.