Question 9
(USA) Let $A_1A_2A_3$ be a nonisosceles triangle with incenter $I$. Let $C_i$, $i=1,2,3$, be the smaller circle through $I$ tangent to $A_iA_{i+1}$ and $A_iA_{i+2}$ (the addition of indices being mod $3$). Let $B_i$, $i=1,2,3$, be the second point of intersection of $C_{i+1}$ and $C_{i+2}$. Prove that the circumcenters of the triangles $A_1B_1I,A_2B_2I,A_3B_3I$ are collinear.