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concurrent wanted, circles with medians as diamaters intersect in pairs

Source: 2009 Oral Moscow Geometry Olympiad grades 10-11 p4

September 14, 2020
geometryconcurrencyconcurrentMediansdiameter

Problem Statement

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let C1C_1 be the intersection point of the circles built on the medians AM1AM_1 and BM2BM_2, which is more distant from the vertex CC. Points A1A_1 and B1B_1 are defined similarly. Prove that the lines AA1,BB1AA_1, BB_1 and CC1CC_1 intersect at one point.
(D. Tereshin)
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