Lambert summation

The Steiner ellipse of an isosceles triangle

In geometry, the Steiner ellipse, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique ellipse that touches a given triangle at its vertices and whose center is the triangle's centroid. 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 ${\frac {4\pi }{3{\sqrt {3}}}},$ 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.

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