A circumellipse is a circumconic of a triangle that is an ellipse. The billiard
circumellipse has its center at the mittenpunkt 
of the triangle
(Reznik and Garcia 2021).
More generally, a circumellipse of a quadrilateral is an ellipse that passes through all four of its vertices. Every convex quadrilateral has a unique circumellipse of minimal area and a unique circumellipse of minimal eccentricity (Horwitz 2010).
The minimum-area circumellipse of a quadrilateral need not be its Löwner ellipsoid. The former minimizes area among ellipses passing through all four vertices, whereas the latter minimizes area among all ellipses containing the quadrilateral and may have only three vertices on its boundary (Horwitz 2010).
A convex quadrilateral is called a bielliptic quadrilateral if its unique minimum-eccentricity inellipse and circumellipse have the same eccentricity (Horwitz 2010).
There is an amazing formula for the area of a circumellipse. Let 
be the length of the chord of
the ellipse through the center
of the ellipse and parallel
to the sideline 
of the reference triangle 
, and similarly define 
and 
. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
(Chakerian 1979, p. 149), where 
is the circumradius of the
reference triangle and 
its area. Explicitly calculating
the chord lengths for a circumellipse
with positive circumconic parameters 
then gives the formula
![A=(4piabcxyzDelta)/([2(abxy+bcyz+cazx)-(a^2x^2+b^2y^2+c^2z^2)]^(3/2))](/images/equations/Circumellipse/NumberedEquation1.svg) |
(3)
|
(E. W. Weisstein, Dec. 4, 2005).
The following table summarizes the areas of some named circumellipses.
| representative | circumellipse | center | area |
| circumcircle |  |  |
| excentral-hexyl ellipse |  | ![4piabc[((a+b)(b+c)(c+a))/(a+b+c)]^(3/2)Deltaproduct_(cyclic)1/(sqrt(f(a,b,c)))](/images/equations/Circumellipse/Inline19.svg) |
| Macbeath circumconic |  |  |
| Steiner circumellipse |  |  |
