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Let ABC be a triangle with bisectors AA_1,BB_1, CC_1 (A_1 \in  BC, etc.) and M their common point. Consider the triangles MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then AB = BC = CA.

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