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orthocenter, incenter of different triangles and midpoint are collinear

Source: 2017 Oral Moscow Geometry Olympiad grades 10-11 p5

July 25, 2019
geometryincentercollinearorthocenter

Problem Statement

The inscribed circle of the non-isosceles triangle ABCABC touches sides AB,BCAB, BC and ACAC at points C1,A1C_1, A_1 and B1B_1, respectively. The circumscribed circle of the triangle A1BC1A_1BC_1 intersects the lines B1A1B_1A_1 and B1C1B_1C_1 at the points A0A_0 and C0C_0, respectively. Prove that the orthocenter of triangle A0BC0A_0BC_0, the center of the inscribed circle of triangle ABCABC and the midpoint of the ACAC lie on one straight line.