Question 15
Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK$ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK), (ACH)$ respectively. Prove the following assertions:
1. If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$.
2. If $XY \parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.
1. If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$.
2. If $XY \parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.