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concurrent, touchpoints of incircle, 3 orthocenters

Source: VI Soros Olympiad 1990-00 R3 9.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
geometryconcurrencyconcurrent

Problem Statement

Let A1,A_1, B1B_1, C1C_1 be the touchpoints of the circle inscribed in the acute triangle ABCABC (A1A_1 is the touchpoint with the side BCBC, etc.). Let A2A_2, B2B_2, C2C_2 be the intersection points of the altitudes of triangles AB1C1AB_1C_1, A1BC1A_1BC_1 and A1B1CA_1B_1C respectively. Prove that the lines A1A2A_1A_2 and B1B2B_1B_2 and C1C2C_1C_2 intersect at one point.
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