Question 145
The incircle of a triangle $A_0B_0C_0$ touches the sides $B_0C_0$, $C_0A_0$, $A_0B_0$ at the points $A$, $B$, $C$, respectively, and the incircle of the triangle $ABC$ with incenter $I$ touches the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Let $\sigma(ABC)$ and $\sigma(A_1B_1C)$ be the areas of the triangles $ABC$ and $A_1B_1C$ respectively. Show that if $\sigma(ABC) = 2\sigma(A_1B_1C)$, then the lines $AA_0$, $BB_0$, $IC_1$ pass through a common point.