The MacBeath triangle 
(a term coined here for the first time), is the triangle whose vertices are the contact
points of the MacBeath inconic with the reference
triangle 
.
It is the polar triangle of the MacBeath inconic.
It has trilinear vertex matrix
![[0 c^2cosCsecB b^2; c^2 0 a^2cosAsecC; b^2cosBsecA a^2 0].](/images/equations/MacBeathTriangle/NumberedEquation1.svg) |
(1)
|
It is perspective with the reference triangle, with perspector (corresponding to the
Brianchon point of the MacBeath
inconic) given by Kimberling center 
, which has equivalent triangle center functions
 |  |  |
(2)
|
 |  |  |
(3)
|
It has area
 |
(4)
|
where 
is the area of the reference triangle.
The circumcircle of the MacBeath triangle is the MacBeath circle.
