CZ-SK-MO 2015 Problem 5

In given triangle $\triangle ABC$ , difference between sizes of each pair of sides is at least $d>0$ . Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$ .