Question 153
Let $ABC$ be a triangle such that the circle $\omega$ with diameter $BC$ is tangent to the exterior bisector of $\angle BAC$. The internal bisector of $\angle BAC$ intersects with the side $BC$ at point $K$ and with the circumcircle of $ABC$ at point $L \neq A$. Let $M$ be the midpoint of $BC$. Prove that the circumcircle of $KLM$ is tangent to $\omega$.