CN-TST 2019 Problem 17

In $\Delta ABC$ , $AD \perp BC$ at $D$ . $E,F$ lie on line $AB$ , such that $BD=BE=BF$ . Let $I,J$ be the incenter and $A$ -excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$ , and $\Delta PEI \sim \Delta QFJ$ .