USAJMO 2016 Problem 1

The isosceles triangle $\triangle ABC$ , with $AB=AC$ , is inscribed in the circle $\omega$ . Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$ , and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$ , respectively.

Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.