Question 12
Let $I$ be the incenter and $\Omega$ be the inscribed circle of a triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $K$ be the point of intersection of the line, going through $A$ and perpendicular to line $BC$, and the line, going through $M$ and perpendicular to the line $AI$. Prove that the circle having the line segment $AK$ as a diameter is tangent to the circle $\Omega$.