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 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.
for side lengths a, b, c.
Axes and foci
The semi-major and semi-minor axes have lengths
and focal length
The foci are called the Bickart points of the triangle.
Given a triangle with vertices
the linear problem
can be solved, and the eigenvalues of the matrix form of the solution
are 3 times the squared inverse lengths of the semi-major axis and semi-minor axis; the corresponding eigenvectors relate to the orientation. This approach generalizes to higher dimensions.
- Weisstein, Eric W. "Steiner Circumellipse." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/SteinerCircumellipse.html