UA-UMO 2023 Problem 103

Let $I$ be the incenter of the triangle $ABC$ , and $P$ be any point on the arc $BAC$ of its circumcircle. Points $K$ and $L$ are chosen on the tangent to the circumcircle $\omega$ of triangle $API$ at point $I$ , so that $BK = KI$ and $CL = LI$ . Show that the circumcircle of triangle $PKL$ is tangent to $\omega$ .

Proposed by Mykhailo Shtandenko