The Steiner circumellipse is the circumellipse that is the isotomic conjugate of the line at infinity and the isogonal conjugate of the Lemoine axis. It has circumconic parameters

(1)

giving trilinear equation

(2)

(Vandeghen 1965; Kimberling 1998, p. 236). The Steiner circumellipse is often simply called the "Steiner ellipse," but the designation "circumellipse" is useful to distinguish it from the less important curve known as the Steiner inellipse.

It is the unique ellipse passing through the vertices of a triangle and having the triangle centroid of as its center. The Steiner circumellipse is the also ellipse of least area that passes through , , and (Kimberling).

The area of the Cevian triangle of any point on the Steiner circumellipse is , where is the area of the reference triangle.

The polar triangle of the Steiner circumellipse is the anticomplementary triangle.

The foci of the Steiner circumellipse are known as the Bickart points. The Steiner circumellipse has semiaxes lengths

			(3)
			(4)

and focal length

(5)

where

(6)

and has area

(7)

where is the area of the reference triangle (P. Moses, pers. comm., Dec. 31, 2004).

The intersections of the major and minor axes with the Steiner circumellipse are given by , where , , and are given by roots of the quartic equations

	(8)
	(9)
	(10)

Explicitly, the intersections with the major axis are

(11)

and the intersections with the minor axis are

(12)

It passes through Kimberling centers for (the Steiner point), 190 (Brianchon point of the Yff parabola), 290, 648, 664, 666, 668, 670, 671, 886, 889, 892, 903, 1121, 1494, 2479, 2480, 2481, and 2966.

The Steiner circumellipse also shares the Steiner point together with the points , , and with the circumcircle of (Kimberling 1998, p. 236; Kimberling).

The minor axis of the ellipse can be constructed as the angle bisector either of or , where is the Steiner point, is the Tarry point, is the circumcenter, and is the symmedian point (Conway 2000). These axes are parallel to the asymptotes of the Kiepert hyperbola (Conway 2000, Yiu 2003).

Another nice construction is to construct the intersections of the Lemoine axis with the Parry circle and then note that joining the triangle centroid with the intersections gives the axes (P. Moses, pers. comm., Dec. 31, 2004).

The fourth intersection of the Steiner circumellipse with a rectangular circumhyperbola for has center function

(13)

which is the isogonal conjugate of the intersection of the line through the circumcenter and the isogonal conjugate of and the Lemoine axis. The following table summarizes these points for various named rectangular circumhyperbolas (P. Moses, pers. comm., Dec. 31, 2004).

circumhyperbola	Kimberling
Feuerbach hyperbola
Jerabek hyperbola
Kiepert hyperbola

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., pp. 451-458, 1893.

Conway, J. H. Message 1237. Hyacinthos mailing list. Aug. 18, 2000.

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 108, 1913.

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Kimberling, C. "Steiner Point." http://faculty.evansville.edu/ck6/tcenters/class/steiner.html.

Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle." Amer. Math. Monthly 72, 1091-1094, 1965.

Yiu, P. "On the Fermat Lines." Forum Geom. 3, 73-81, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200307index.html.

CITE THIS AS:

Weisstein, Eric W. "Steiner Circumellipse." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SteinerCircumellipse.html