Question 182
Let $I$ be the incentre of triangle $A B C$ with $A B \gt A C$ and let the line $A I$ intersect the side $B C$ at $D$. Suppose that point $P$ lies on the segment $B C$ and satisfies $P I = P D$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $B C$, and let $Q$ be the other intersection of the circumcircles of the triangles $A B C$ and $A P D$. Prove that $\angle B A Q = \angle C A J$.