The Medians of a Triangle Divide It into Three Smaller Triangles of Equal Area

The segments drawn from the centroid of a triangle to each of its vertices divide the triangle into three smaller triangles that have the same area. A visual proof is given here.

Euclid I.37 states that triangles with the same base and height have the same area. The original language was something like, "Triangles which are on the same base and in the same parallels equal one another." See Euclid's Elements, Book I, Proposition 37.