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inscribed equilateral, midpoints, tangents, intersections

Source: VI All-Ukrainian Tournament of Young Mathematicians, Qualifying p2

May 25, 2021
geometryEquilateralUkrainian TYM

Problem Statement

Let A1,B1,C1A_1,B_1,C_1 be the midpoints of the sides of the BC,AC,ABBC,AC, AB of an equilateral triangle ABCABC. Around the triangle A1B1C1A_1B_1C_1 is a circle γ\gamma, to which the tangents B2C2B_2C_2, A2C2A_2C_2, A2B2A_2B_2 are drawn, respectively, parallel to the sides BC,AC,ABBC, AC, AB. These tangents have no points in common with the interior of triangle ABCABC. Find out the mutual location of the points of intersection of the lines AA2AA_2 and BB2BB_2, AA2AA_2 and CC2CC_2, BB2BB_2 and CC2CC_2 and the circumscribed circle γ\gamma. Try to consider the case of arbitrary points A1,B1,C1A_1,B_1,C_1 located on the sides of the triangle ABCABC.
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