RO-NMO 2000 Problem 94

Let $I$ be the center of the incircle of a triangle $ABC.$ Shw that, if for any point $M$ on the segment $AB$ (extremities excluded) there exist two points $N,P$ on $BC,$ respectively, $AC$ (both excluding the extremities) such that the center of mass of $MNP$ coincides with $I,$ then $ABC$ is equilateral.