Question 42
Given a triangle $ABC$, the line parallel to the side $BC$ and tangent to the incircle of the triangle meets the sides $AB$ and $AC$ at the points $A_1$ and $A_2$; the points $B_1$, $B_2$ and $C_1$, $C_2$ are defined similarly. Show that
$$
AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \geq \frac{1}{9} (AB^2 + BC^2 + CA^2)
$$
and determine the cases of equality.
$$
AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \geq \frac{1}{9} (AB^2 + BC^2 + CA^2)
$$
and determine the cases of equality.