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Prove that at least two triangle are congruent

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September 22, 2010
geometrycircleTrianglecongruent trianglesIMO Shortlist

Problem Statement

Let Ar,Br,CrA_r,B_r, C_r be points on the circumference of a given circle SS. From the triangle ArBrCrA_rB_rC_r, called Δr\Delta_r, the triangle Δr+1\Delta_{r+1} is obtained by constructing the points Ar+1,Br+1,Cr+1A_{r+1},B_{r+1}, C_{r+1} on SS such that Ar+1ArA_{r+1}A_r is parallel to BrCrB_rC_r, Br+1BrB_{r+1}B_r is parallel to CrArC_rA_r, and Cr+1CrC_{r+1}C_r is parallel to ArBrA_rB_r. Each angle of Δ1\Delta_1 is an integer number of degrees and those integers are not multiples of 4545. Prove that at least two of the triangles Δ1,Δ2,…,Δ15\Delta_1,\Delta_2, \ldots ,\Delta_{15} are congruent.
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