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point on concurrency of 3 lines lies on circumcircle, // passing from midpoints

Source: Sharygin 2005 finals 11.1

August 30, 2019
concurrentconcurrencyparallelmidpointsgeometrycircumcircle

Problem Statement

A1,B1,C1A_1, B_1, C_1 are the midpoints of the sides BC,CA,BABC,CA,BA respectively of an equilateral triangle ABCABC. Three parallel lines, passing through A1,B1,C1A_1, B_1, C_1 intersect, respectively, lines B1C1,C1A1,A1B1B_1C_1, C_1A_1, A_1B_1 at points A2,B2,C2A_2, B_2, C_2. Prove that the lines AA2,BB2,CC2AA_2, BB_2, CC_2 intersect at one point lying on the circle circumscribed around the triangle ABCABC.
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