Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral

Theorem
Let $\triangle ABC$ be a triangle.

Let $O$ be the circumcenter of $\triangle ABC$.

Let $G$ be the centroid of $\triangle ABC$.

Let $H$ be the orthocenter of $\triangle ABC$.

Then $O$, $G$ and $H$ are the same points $\triangle ABC$ is equilateral.

If $\triangle ABC$ is not equilateral, then $O$, $G$ and $H$ are all distinct.

Necessary Condition
Let $\triangle ABC$ be an equilateral triangle.