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
(1)
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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)
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(3)
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It has area
(4)
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where is the area of the reference triangle.
The circumcircle of the MacBeath triangle is the MacBeath circle.