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Polar bisects segment

Source: Sharygin Geometry Olympiad, Final Round 2016, Problem 5 grade 9

August 4, 2016
geometryangles

Problem Statement

The center of a circle ω2\omega_2 lies on a circle ω1\omega_1. Tangents XPXP and XQXQ to ω2\omega_2 from an arbitrary point XX of ω1\omega_1 (PP and QQ are the touching points) meet ω1\omega_1 for the second time at points RR and SS. Prove that the line PQPQ bisects the segment RSRS.
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