TH-TMO 2021 Problem 1

Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$ . Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$ . Draw a tangent from $B$ to $\omega$ at $P$ , and draw a tangent from $C$ to $\omega$ at $Q$ . Suppose that the line $PQ$ intersects the line $BC$ at point $M$ . Prove that $M$ is the midpoint of $BC$ .