Question 84
Points $A_{1}, B_{1}, C_{1}$ lie on the sides $BC$, $AC$ and $AB$ of a triangle $ABC$, respectively, such that $AB_{1} - AC_{1} = CA_{1} - CB_{1} = BC_{1} - BA_{1}$. Let $I_{A}, I_{B}, I_{C}$ be the incenters of triangles $AB_{1}C_{1}$, $A_{1}BC_{1}$ and $A_{1}B_{1}C$ respectively. Prove that the circumcenter of triangle $I_{A}I_{B}I_{C}$ is the incenter of triangle $ABC$.