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similar triangles wanted, cevians, circumcircle, symmetric points wrt sides

Source: 2006 Oral Moscow Geometry Olympiad grades 10-11 p6

October 20, 2020
geometrysimilar trianglescircumcircleCevians

Problem Statement

Given triangle ABCABC and points PP. Let A1,B1,C1A_1,B_1,C_1 be the second points of intersection of straight lines AP,BP,CPAP, BP, CP with the circumscribed circle of ABCABC. Let points A2,B2,C2A_2, B_2, C_2 be symmetric to A1,B1,C1A_1,B_1,C_1 wrt BC,CA,ABBC,CA,AB, respectively. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are similar.
(A. Zaslavsky)
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