The Macbeath triangle Δ X_A X_B X_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 Δ A B C. It is the polar triangle of the Macbeath inconic. It has trilinear vertex matrix [0 | c^2 cos C sec B | b^2 c^2 | 0 | a^2 cos A sec C b^2 cos B sec A | a^2 | 0].

Macbeath circle | Macbeath inconic

Alexander Murray Macbeath