MacBeath Triangle


The MacBeath triangle DeltaX_AX_BX_C (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 DeltaABC.

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].

It is perspective with the reference triangle, with perspector (corresponding to the Brianchon point of the MacBeath inconic) given by Kimberling center X_(264), which has equivalent triangle center functions


It has area


where Delta is the area of the reference triangle.

The circumcircle of the MacBeath triangle is the MacBeath circle.

See also

MacBeath Inconic, MacBeath Circle

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Cite this as:

Weisstein, Eric W. "MacBeath Triangle." From MathWorld--A Wolfram Web Resource.

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