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Three circles implies incenter

Source: Brazil National Olympiad 2019 #1

November 14, 2019
geometryincenter

Problem Statement

Let ω1\omega_1 and ω2\omega_2 be two circles with centers C1C_1 and C2C_2, respectively, which intersect at two points PP and QQ. Suppose that the circumcircle of triangle PC1C2PC_1C_2 intersects ω1\omega_1 at A≠PA \neq P and ω2\omega_2 at B≠PB \neq P. Suppose further that QQ is inside the triangle PABPAB. Show that QQ is the incenter of triangle PABPAB.
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