An unusual proof of the centroid theorem

In this article, I present an unusual proof of the Centroid Theorem. (The theorem states: For any triangle, the three medians meet in a point. Moreover, the common point of intersection is a point of trisection of each median.) The standard methods (see [3], pg 65 for a much shorter proof that uses the same base results as this one, or [1], pg 7 for one that uses Ceva’s theorem) require nothing but elementary geometry. Another vector-based approach (see [2], pg 19) also exists. This one, however, makes use of an infinite geometric progression to achieve its result