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prove that the radical center of omega_2, omega_3, omega_4 is in the ratio 2:1

Source: Sharygin Finals 2023 10.8

August 2, 2023
geometrySharygin Geometry OlympiadSharygin 2023

Problem Statement

A triangle ABCABC is given. Let ω1\omega_1, ω2\omega_2, ω3\omega_3, ω4\omega_4 be circles centered at points XX, YY, ZZ, TT respectively such that each of lines BCBC, CACA, ABAB cuts off on them four equal chords. Prove that the centroid of ABCABC divides the segment joining XX and the radical center of ω2\omega_2, ω3\omega_3, ω4\omega_4 in the ratio 2:12:1 from XX.
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