Question 87
Let $ABC$ be a triangle, $H_{a}, H_{b}$ and $H_{c}$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_{a}, T_{b}, T_{c}$ its intouch points on the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $A H_{b} H_{c}$ and $A T_{b} T_{c}$ intersect again at $A'$. The circumcircles of triangles $B H_{c} H_{a}$ and $B T_{c} T_{a}$ intersect again at $B'$. The circumcircles of triangles $C H_{a} H_{b}$ and $C T_{a} T_{b}$ intersect again at $C'$. Prove that the points $A', B', C'$ are collinear.