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2012 ToT Fall Junior A p4 incenters coincide, equilateral criterion?

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March 22, 2020
geometryincenterEquilateral

Problem Statement

Given a triangle ABCABC. Suppose I is its incentre, and X,Y,ZX, Y, Z are the incentres of triangles AIB,BICAIB, BIC and AICAIC respectively. The incentre of triangle XYZXYZ coincides with II. Is it necessarily true that triangle ABCABC is regular?
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