KURSCHAK 2004 Problem 1

Given is a triangle $ABC$ , its circumcircle $\omega$ , and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$ , respectively. Prove that the $A$ -excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$ .