The MacBeath inconic of a triangle is the inconic with parameters
(1)
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Its foci are the circumcenter and the orthocenter , giving the center as the nine-point center .
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 , which is Kimberling center and has triangle center function
(2)
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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)
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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.