Question 26
Let $ABC$ be a triangle, $I$ its in-centre; $A_{1}, B_{1}, C_{1}$ be the reflections of $I$ in $BC, CA, AB$ respectively. Suppose the circum-circle of triangle $A_{1}B_{1}C_{1}$ passes through $A$. Prove that $B_{1}, C_{1}, I, I_{1}$ are concyclic, where $I_{1}$ is the in-centre of triangle $A_{1}B_{1}C_{1}$.