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beautiful geometry problem.

Source: Iranian second round 2020 day1 P3

July 14, 2020
geometry

Problem Statement

let ω1\omega_1 be a circle with O1O_1 as its center , let ω2\omega_2 be a circle passing through O1O_1 with center O2O_2 let AA be one of the intersection of ω1\omega_1 and ω2\omega_2 let xx be a line tangent line to ω1\omega_1 passing from AA let ω3\omega_3 be a circle passing through O1,O2O_1,O_2 with its center on the line xx and intersect ω2\omega_2 at PP (not O1O_1) prove that the reflection of PP through xx is on ω1\omega_1
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