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Prove that AB = BC = CA - ISL 1976

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September 20, 2010
geometrybisectorincircleIMO Shortlist

Problem Statement

Let ABCABC be a triangle with bisectors AA1,BB1,CC1AA_1,BB_1, CC_1 (A1∈BCA_1 \in BC, etc.) and MM their common point. Consider the triangles MB1A,MC1A,MC1B,MA1B,MA1C,MB1CMB_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.AB = BC = CA.
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