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MacBeath Inconic


MacBeathInconic

The MacBeath inconic of a triangle is the inconic with parameters

 x:y:z=a^2cosA:b^2cosB:c^2cosC.
(1)

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

It is named after Macbeath (1951), but had earlier been investigated by Serret (1865). It was subsequently publicized by Gabriel-Marie (1912).

The Brianchon point is the isotomic conjugate of the circumcenter O, which is Kimberling center X_(264) and has triangle center function

 alpha_(264)=(secA)/(a^2).
(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

 A=(a^2b^2c^2pisqrt(cosAcosBcosC))/(16sqrt(2)Delta^2),
(3)

where Delta is the area of the reference triangle.

The MacBeath inconic passes through Kimberling centers X_i for i=339, 1312, 1313, 2968, 2969, 2970, 2971, 2972, 2973, and 2974.

P. Moses (Nov. 12, 2004) noted that if a point X lies on this conic, then the reflections of X in X_5 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.


See also

Inconic, MacBeath Circumconic, MacBeath Triangle

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References

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.aloj.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.

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MacBeath Inconic

Cite this as:

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

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