Draw triangle ABC with midpoints F, D and E.

Using the crossbar theorem, show point G is the intersection between two angles.

Draw the midpoints of AG and CG.

Find similar triangles FBD and ABC; and AGC and hiG, then congruent triangles hDi, and FDh.

Draw parallelogram FhiD. The diagonal properties of a parallelogram state they bisect each other. If two different verteces were choosen instead of the original chosen, you will find the same intersection of G.

and because point G is inside each angle,G is in the interior of triangle

ABC, proving point G is the centroid.