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Fixed point as P varies

Source: 2016 USAJMO 1

April 19, 2016
USAJMOgeometry

Problem Statement

The isosceles triangle △ABC\triangle ABC, with AB=ACAB=AC, is inscribed in the circle ω\omega. Let PP be a variable point on the arc BC⌢\stackrel{\frown}{BC} that does not contain AA, and let IBI_B and ICI_C denote the incenters of triangles △ABP\triangle ABP and △ACP\triangle ACP, respectively.
Prove that as PP varies, the circumcircle of triangle △PIBIC\triangle PI_BI_C passes through a fixed point.
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