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Let O be an interior point of acute triangle ABC. Let A_1 lie on BC with OA_1 perpendicular to BC. Define B_1 on CA and C_1 on AB similarly. Prove that O is the circumcenter of ABC if and only if the perimeter of A_1B_1C_1 is not less than any one of the perimeters of AB_1C_1, BC_1A_1, and CA_1B_1.

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