Medians of Triangle Meet at Centroid

Theorem
Let $ABC$ be a triangle.

Then the medians of $ABC$ meet at a single point, which is the centroid.

Proof
Let $\vec a, \vec b, \vec c$ be $\vec{OA}, \vec{OB}, \vec{OC}$ respectively.

Let the midpoint of $BC, AC, AB$ be $\vec d, \vec e, \vec f$ respectively.

Then:

The three medians are $\vec{AD}, \vec{BE}, \vec{CF}$ respectively:

Their equations:

It can be verified that $x = y = z = \dfrac 2 3$ produce the same point.