Question 52
Suppose that $I$ is the incenter of triangle $ABC$. The perpendicular to line $AI$ from point $I$ intersects sides $AC$ and $AB$ in points $B'$ and $C'$, respectively. Points $B_1$ and $C_1$ are placed on half-lines $BC$ and $CB$, respectively, in such a way that $AB = BB_1$ and $AC = CC_1$. If $T$ is the second intersection point of the circumcircles of triangles $AB_1C'$ and $AC_1B'$, prove that the circumcenter of triangle $ATI$ lies on the line $BC$.
