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equal segments starting with 2 intersecting circles

Source: Iranian Geometry Olympiad 2018 IGO Advanced p1

September 19, 2018
geometrycirclesequal segments

Problem Statement

Two circles ω1,ω2\omega_1,\omega_2 intersect each other at points A,BA,B. Let PQPQ be a common tangent line of these two circles with P∈ω1P \in \omega_1 and Q∈ω2Q \in \omega_2. An arbitrary point XX lies on ω1\omega_1. Line AXAX intersects ω2 \omega_2 for the second time at YY . Point Y′≠YY'\ne Y lies on ω2\omega_2 such that QY=QY′QY = QY'. Line Y′BY'B intersects ω1 \omega_1 for the second time at X′X'. Prove that PX=PX′PX = PX'.
Proposed by Morteza Saghafian
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