How to prove that there is no other point except centroid which can divides in n three parts of equal area?

Question

We know that centroid divides the triangle in three parts of equal area. My question is how to prove that there is no other point except centroid which divides in that manner.

Answer 1

I would do it by constructing it, starting from the conditions of equal areas. Draw the heights from two vertices. Take $1/3^{rd}$ of the heights from the opposing base, and draw a parallels to those bases. The point that forms a triangle with area $1/3^{rd}$ of the original triangle must be on each of these lines. They intersect in only one point. So there is only a single point with that property. You show (by other means) that the centroid has this property, so than the centroid is the only point dividing the triangle in three equal parts.