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Nine-point circle

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September 1, 2010
geometrycircumcirclerhombusparallelogramIMO ShortlistIMO Longlist

Problem Statement

Let ABCABC be a triangle, OO its circumcenter, SS its centroid, and HH its orthocenter. Denote by A1,B1A_1, B_1, and C1C_1 the centers of the circles circumscribed about the triangles CHB,CHACHB, CHA, and AHBAHB, respectively. Prove that the triangle ABCABC is congruent to the triangle A1B1C1A_1B_1C_1 and that the nine-point circle of △ABC\triangle ABC is also the nine-point circle of △A1B1C1\triangle A_1B_1C_1.
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