Question 94
A triangle $ABC$ is given every two sides of which differ in length by at least $d \gt 0$. Denote by $T$ its centroid, $I$ incentre and $\rho$ inradius. Prove that
$$
S_{AIT} + S_{BIT} + S_{CIT} \geq \frac{2}{3} \rho d,
$$
where $S_{XYZ}$ denotes the area of triangle $XYZ$.
$$
S_{AIT} + S_{BIT} + S_{CIT} \geq \frac{2}{3} \rho d,
$$
where $S_{XYZ}$ denotes the area of triangle $XYZ$.