The MacBeath inconic of a triangle is the inconic with parameters

(1)

Its foci are the circumcenter and the orthocenter , giving the center as the nine-point center .

It is named after Macbeath (1951), who showed that this conic is the inconic enclosing maximum area. However, it had earlier been investigated by Serret (1865) and subsequently publicized in Gabriel-Marie (1912).

The Brianchon point is the isotomic conjugate of the circumcenter , which is Kimberling center and has triangle center function

(2)

The triangle formed by the contact points of the MacBeath inconic with the reference triangle is called the MacBeath triangle.

The polar triangle of the MacBeath inconic is the MacBeath triangle.

When the MacBeath inconic is an inellipse, it has area

(3)

where is the area of the reference triangle.

The MacBeath inconic passes through Kimberling centers for , 1312, 1313, 2968, 2969, 2970, 2971, 2972, 2973, and 2974.

P. Moses (Nov. 12, 2004) noted that if a point lies on this conic, then the reflections of in and in the Euler line lie on the conic.

The MacBeath inconic is traditionally called the "Macbeath inellipse," although it is an ellipse only for acute triangles. For obtuse triangles, it is a hyperbola.

Brisse, E. "Table of Centers on Named Objects in Triangle Geometry of Degree 1-2-3-4." http://pages.infinit.net/spqrsncf/ngorecent.htm#L2I-11.

Gabriel-Marie, F. Problem 130 in Exercices de géométrie, comprenant l'esposé des méthodes géométriques et 2000 questions résolues, 5th ed. Tours, France: Maison Mame, 1912.

García Capitán, F. J. "Sobre la ellipse inscrita OH." http://www.personal.us.es/rbarroso/eipseOH/indice.htm.

Macbeath, A. M. "A Compactness Theorem for Affine Equivalence-Classes of Convex Regions." Canad. J. Math. 3, 54-61, 1951.

Seret, P. Nouv. Ann. Math., p. 428, 1865.

CITE THIS AS:

Weisstein, Eric W. "MacBeath Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MacBeathInconic.html