TSTST 2019 Problem 9

Let $A B C$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $B C$ such that the incircles of $\triangle A B K$ and $\triangle A B L$ are tangent at $P$, and the incircles of $\triangle A C K$ and $\triangle A C L$ are tangent at $Q$. Prove that $I P=I Q$.