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concurrent circles passing through midpoints of triangle's sides (orthocenter)

Source: Sharygin 2006 X-XI CR 20

August 26, 2019
orthocentergeometrycirclesmidpointconcurrencyconcurrent

Problem Statement

Four points are given A,B,C,DA, B, C, D. Points A1,B1,C1,D1A_1, B_1, C_1,D_1 are orthocenters of the triangles BCD,CDA,DAB,ABCBCD, CDA, DAB, ABC and A2,B2,C2,D2A_2, B_2, C_2,D_2 are orthocenters of the triangles B1C1D1,C1D1A1,D1A1B1,A1B1C1B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1 etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
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