On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.
Problem
Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade
Tags: geometry, Inscribed circle, circumcircle