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excircles touch corresponding sides of triangle ABC

Source: ARO 2005 - problem 11.3

April 30, 2005
geometrycircumcirclegeometry solved

Problem Statement

Let A′, B′, C′A',\,B',\,C' be points, in which excircles touch corresponding sides of triangle ABCABC. Circumcircles of triangles A′B′C, AB′C′, A′BC′A'B'C,\,AB'C',\,A'BC' intersect a circumcircle of ABCABC in points C1≠C, A1≠A, B1≠BC_1\ne C,\,A_1\ne A,\,B_1\ne B respectively. Prove that a triangle A1B1C1A_1B_1C_1 is similar to a triangle, formed by points, in which incircle of ABCABC touches its sides.
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