Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?
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Tags: geometry, incenter, Equilateral
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?