Do the the circumcenter, the orthocenter, and the centroid ever coincide?

The Euler line passes through the circumcenter, the orthocenter, and the centroid. Can it ever happen that these three coincide? In the interactive sketch below, the circumcenter, the orthocenter, and the centroid of a triangle are shown along with the Euler line. Drag any of the vertices of the triangle to see how close you can make the circumcenter, orthocenter, and the centroid to each other. Do they ever coincide? The side lengths are given for the triangle to help you characterize triangles whose circumcenter, orthocenter, and the centroid are coincident.

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Theorem

The circumcenter, orthocenter, and the centroid of a triangle coincide if and only if the triangle is equilateral.

How are the distances between the circumcenter, the orthocenter, and the centroid related?

In the interactive sketch below, the distance from the centroid to the orthocenter, the distance from the circumcenter to the orthocenter, and the ratio of these two distances are given. How are the distances related to each other? What can you say about the ratio of the distance from the centroid to the circumcenter and the distance from the centroid to the orthocenter? Drag any of the vertices of the triangle to change the triangle.

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Theorem

The centroid is one third of the way from the circumcenter to the orthocenter.

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