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prove that all the points X are collinear as radius R changes, 3 circles related

Source: Sharygin 2013 Final 9.7

August 19, 2018
geometryLocuscollinearcircles

Problem Statement

Two fixed circles ω1\omega_1 and ω2\omega_2 pass through point OO. A circle of an arbitrary radius RR centered at OO meets ω1\omega_1 at points AA and BB, and meets ω2\omega_2 at points CC and DD. Let XX be the common point of lines ACAC and BDBD. Prove that all the points X are collinear as RR changes.
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