USAJMO 2016 Question 1

The isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $P$ vary on the arc $BC$ not containing $A$, and let $I_B,I_C$ be the incenters of $ABP,ACP$. Prove that the circumcircle of $PI_BI_C$ passes through a fixed point.

Proposed by Ivan Borsenco, Zuming Feng.