A conic section that is tangent to all sides of a triangle is called an inconic. Any trilinear equation of the form
 |
(1)
|
where 
, 
, and 
are functions of the side lengths 
, 
,
and 
, 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 (Veblen and Young 1938, p. 111; Eddy and Fritsch 1994).
The inconic parameters are given simply in terms of the trilinear coordinates 
of the Brianchon point
as
 |
(2)
|
Furthermore, the center of an inconic with parameters 
is the point
 |
(3)
|
(Kimberling 1998, p. 238).
An inconic is a parabola iff
 |
(4)
|
in which case the focus is the point 
which lies on the circumcircle
and the directrix passes through the orthocenter
(Smith 1894, p. 70; Eddy and Fritsch 1994; Kimberling 1998, p. 239).
Examples of inconics include the Brocard inellipse, incircle, Kiepert parabola, Steiner inellipse, and Yff parabola.
