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3 parallel lines intersecting circumecirle, reflections, and lead to concurrent

Source: 2012 Sharygin Geometry Olympiad Final Round 9.2

August 3, 2018
geometrygeometric transformationreflectionconcurrencyconcurrent

Problem Statement

Three parallel lines passing through the vertices A,BA, B, and CC of triangle ABCABC meet its circumcircle again at points A1,B1A_1, B_1, and C1C_1 respectively. Points A2,B2A_2, B_2, and C2C_2 are the reflections of points A1,B1A_1, B_1, and C1C_1 in BC,CABC, CA, and ABAB respectively. Prove that the lines AA2,BB2,CC2AA_2, BB_2, CC_2 are concurrent.
(D.Shvetsov, A.Zaslavsky)
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