Steiner ellipse

Not to be confused with Steiner conic.

In geometry, the Steiner ellipse of a triangle is the unique circumellipse (an ellipse that touches the triangle at its vertices) whose center is the triangle's centroid. It is also called the Steiner circumellipse, to distinguish it from the Steiner inellipse. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.

The area of the Steiner ellipse equals the area of the triangle times and hence is 4 times the area of the Steiner inellipse. The Steiner ellipse has the least area of any ellipse circumscribed about the triangle.

The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid). Hence both ellipses are similar (have the same eccentricity).

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