IGO 2021 A5

Given a triangle $ABC$ with incenter $I$ . The incircle of triangle $ABC$ is tangent to $BC$ at $D$ . Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$ , respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$ , respectively. Prove that $AD$ is the Euler line of triangle $IKL$ .

Proposed by Le Viet An, Vietnam