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concurrent lines related to reflections of a line wrt to sides of given triangle

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P16

April 6, 2019
geometryreflectionconcurrencyconcurrentReflections

Problem Statement

Given are triangle ABCABC and line \ell. The reflections of \ell in ABAB and ACAC meet at point A1A_1. Points B1,C1B_1, C_1 are defined similarly. Prove that a) lines AA1,BB1,CC1AA_1, BB_1, CC_1 concur, b) their common point lies on the circumcircle of ABCABC c) two points constructed in this way for two perpendicular lines are opposite.