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 $H, N$). Prove the following assertions: a) If $A$ is the center of the circle $(IMN)$, then $BC$ is tangent to $(IMN)$. b) If $I$ is the midpoint of $BC$, then $BC$ is equal to $4$ times of the distance between the centers of two circles $(ABK)$ and $(ACH)$.