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circumcircle of XYZ passes through fixed point, <XPY = 2<BAC

Source: 2020 Czech-Polish-Slovak Match p6

October 8, 2020
geometryequal anglesfixedFixed point

Problem Statement

Let ABCABC be an acute triangle. Let PP be a point such that PBPB and PCPC are tangent to circumcircle of ABCABC. Let XX and YY be variable points on ABAB and ACAC, respectively, such that ∠XPY=2∠BAC\angle XPY = 2\angle BAC and PP lies in the interior of triangle AXYAXY. Let ZZ be the reflection of AA across XYXY. Prove that the circumcircle of XYZXYZ passes through a fixed point.
(Dominik Burek, Poland)
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