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), 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)
|
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.
