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Prove that lines are parallel

Source: Sharygin contest 2008. The correspondence round. Problem 12

September 3, 2008
geometry proposedgeometry

Problem Statement

(A.Myakishev, 9--10) Given a triangle ABC ABC. Point A1 A_1 is chosen on the ray BA BA so that segments BA1 BA_1 and BC BC are equal. Point A2 A_2 is chosen on the ray CA CA so that segments CA2 CA_2 and BC BC are equal. Points B1 B_1, B2 B_2 and C1 C_1, C2 C_2 are chosen similarly. Prove that lines A1A2 A_1A_2, B1B2 B_1B_2, C1C2 C_1C_2 are parallel.
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