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Orthocenter, midpoints, incircle

Source: 9.1 of XX Geometrical Olympiad in honour of I.F.Sharygin

August 6, 2024
geometrygeo

Problem Statement

Let HH be the orthocenter of an acute-angled triangle ABCABC; A1,B1,C1A_1, B_1, C_1 be the touching points of the incircle with BC,CA,ABBC, CA, AB respectively; EA,EB,ECE_A, E_B, E_C be the midpoints of AH,BH,CHAH, BH, CH respectively. The circle centered at EAE_A and passing through AA meets for the second time the bisector of angle AA at A2A_2; points B2,C2B_2, C_2 are defined similarly. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are similar.
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