Question 85
Let $k$ be the incircle of a $\triangle ABC$. A line, parallel to $BC$ touches $k$ and intersects the sides $AB$ and $AC$ at points $A_1$ and $A_2$. Define the points $B_1, B_2$ and $C_1, C_2$ in a similar way. Prove that
$$
9(\overline{AA_1} \cdot \overline{AA_2} + \overline{BB_1} \cdot \overline{BB_2} + \overline{CC_1} \cdot \overline{CC_2}) \ge \overline{AB}^2 + \overline{BC}^2 + \overline{CA}^2.
$$
$$
9(\overline{AA_1} \cdot \overline{AA_2} + \overline{BB_1} \cdot \overline{BB_2} + \overline{CC_1} \cdot \overline{CC_2}) \ge \overline{AB}^2 + \overline{BC}^2 + \overline{CA}^2.
$$