Steiner circumellipse

In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (ellipse that touches the triangle at its vertices) 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 ${\displaystyle {\frac {4\pi }{3{\sqrt {3}}}},}$ 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.

The equation of the Steiner circumellipse in trilinear coordinates is

for side lengths a, b, c.

The semi-major and semi-minor axes have lengths

and focal length

where

The foci are called the Bickart points of the triangle.

Given a triangle with vertices

the linear problem

can be solved, and the reciprocals of the eigenvalues of the matrix form of the solution

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