Question 6
The incircle of an acute triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $A_1$, $B_1$ and $C_1$. Let $K_A$, $K_B$ and $K_C$ be the incircles of the triangles $AB_1C_1$, $A_1BC_1$ and $A_1B_1C$.
Let $t_A$ denote the common tangent of $K_B$ and $K_C$ which intersects the segments $AB$ and $AC$ but not the segment $BC$. Let $t_B$ be the common tangent of $K_A$ and $K_C$ which intersects the segments $AB$ and $BC$ but not the segment $AC$ and let $t_C$ be the common tangent of $K_A$ and $K_B$ which intersects the segments $AC$ and $BC$ but not the segment $AB$.
Prove that the lines $t_A$, $t_B$ and $t_C$ intersect at a single point.
Let $t_A$ denote the common tangent of $K_B$ and $K_C$ which intersects the segments $AB$ and $AC$ but not the segment $BC$. Let $t_B$ be the common tangent of $K_A$ and $K_C$ which intersects the segments $AB$ and $BC$ but not the segment $AC$ and let $t_C$ be the common tangent of $K_A$ and $K_B$ which intersects the segments $AC$ and $BC$ but not the segment $AB$.
Prove that the lines $t_A$, $t_B$ and $t_C$ intersect at a single point.