conic

A conic section that is tangent to all sides of a triangle is called an inconic. Any trilinear equation of the form x^2 α^2 + y^2 β^2 + z^2 γ^2 - 2y z βγ - 2z x γα - 2x y αβ = 0, where x, y, and z are functions of the side lengths a, b, and c, is an inconic, and every inconic has such an equation. The lines connecting the vertices of a triangle and the corresponding contact points of an inconic are concurrent in a point known as the Brianchon point of the inconic. The inconic parameters are given simply in terms of the trilinear coordinates α:β:γ of the Brianchon point as x:y:z = 1/α :1/β :1/γ.

Brianchon point | Brocard inellipse | circumconic | conic section | incircle | Kiepert parabola | Lemoine inellipse | MacBeath inconic | Mandart inellipse | orthic inconic | Steiner inellipse | Yff parabola