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PQR is equilateral - [IMO LongList 1971]

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January 1, 2011
analytic geometrygeometry unsolvedgeometry

Problem Statement

Let ABC,AA1A2,BB1B2,CC1C2ABC,AA_1A_2,BB_1B_2, CC_1C_2 be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments A2B1,B2C1,C2A1A_2B_1,B_2C_1, C_2A_1 by P,Q,RP,Q,R in this order. Prove that the triangle PQRPQR is equilateral.
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