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concurrent lines out of reflections related to touchpoints of incircles

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P8

April 2, 2019
geometryincircleconcurrencyReflectionsreflection

Problem Statement

The incircle of right-angled triangle ABCABC (∠B=90o\angle B = 90^o) touches AB,BC,CAAB,BC,CA at points C1,A1,B1C_1,A_1,B_1 respectively. Points A2,C2A_2, C_2 are the reflections of B1B_1 in lines BC,ABBC, AB respectively. Prove that lines A1A2A_1A_2 and C1C2C_1C_2 meet on the median of triangle ABCABC.
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