I guess by centroid you mean "centre of mass", as opposed to its meaning of "intersection of the medians of a triangle"
1. It's easy to show that the centre of mass must lie on the medians of the triangle, so convince yourself of this first.
2. Find vector equations describing the medians of the triangle, then show that they a) intersect at a common point and b) this common point lies 2/3 of the way from the vertices along the medians.