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given 3 concurrent circles and 3 collinear intersections, equal triangles wanted

Source: 2007 Sharygin Geometry Olympiad Correspondence Round P8

April 25, 2019
geometryconcurrentcollinearcongruent triangles

Problem Statement

Three circles pass through a point PP, and the second points of their intersection A,B,CA, B, C lie on a straight line. Let A1B1,C1A_1 B_1, C_1 be the second meets of lines AP,BP,CPAP, BP, CP with the corresponding circles. Let C2C_2 be the intersections of lines AB1AB_1 and BA1BA_1. Let A2,B2A_2, B_2 be defined similarly. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are equal,
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