Question 119
Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes belonging to the vertices $A$, $B$ and $C$, respectively, of an acute triangle $ABC$, and let $O_A$, $O_B$ and $O_C$ be the incenters of the triangles $AB_1C_1$, $BC_1A_1$ and $CA_1B_1$, respectively. Let $T_A$, $T_B$ and $T_C$ be the points of tangency of the incircle of the triangle $ABC$ to the sides $BC$, $CA$ and $AB$, respectively. Show that the hexagon $T_A O_C T_B O_A T_C O_B$ is equilateral.