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Geometry · Equilateral triangle · Concyclic points · Inscribed angle · Parallel lines · Cyclic quadrilateral

Problem 3, 2001

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RegionalProof

Two equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) lying on the same side of the line \(CR\). Prove that the lines \(AP\) and \(BQ\) are parallel.

A B C R P Q
The dashed segment is \(CR\); the triangles lie on opposite sides of it in pairs, \(A\) with \(P\) and \(B\) with \(Q\).

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Serbian Regional Competition (Okruzno takmicenje) 2001, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source

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