An inconic is a nondegenerate conic section tangent to the three sidelines of a triangle. Tangency may occur on a side or its extension. Its equation in trilinear coordinates has the form
 |
(1)
|
where 
, 
, and 
are nonzero functions of the side lengths 
, 
, and 
.
Conversely, every such equation defines an inconic.
The lines connecting the vertices of the triangle to the corresponding contact points
of an inconic concur at 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
in terms of the trilinear coordinates 
of the Brianchon
point as
 |
(2)
|
For an inconic that is a central conic, with parameters 
, the center is the point
 |
(3)
|
(Kimberling 1998, p. 238).
 |
(4)
|
in which case its focus lies on the circumcircle and its conic section directrix passes through the orthocenter (Smith 1894, p. 70; Eddy and Fritsch 1994; Kimberling 1998, p. 239).
The following table compares some named inconics. The notation 
denotes a Kimberling center;
for the parabolas, the focus
is listed because there is no finite center. The drawings
show nondegenerate examples for particular reference
triangles and are independently scaled. Special triangles
can cause a named construction to become degenerate. In the drawings, the inconic
is blue and its contact points on the lines 
, 
, and 
are 
, 
,
and 
, respectively. The green lines 
, 
, and 
concur at the green Brianchon
point 
in every row, including the parabolas. The red point
marks the center, or the focus 
for a parabola.
For the Steiner inellipse, 
and the center coincide at the
triangle centroid 
,
so the red marker lies inside the larger green marker. Only finite portions of the
parabolas are shown.
| representative | inconic | center or focus | distinguishing property |
| Brocard inellipse | center: Brocard midpoint  | foci at the Brocard points |
| incircle | center: incenter  | inside the triangle and tangent to all three sides |
| Kiepert parabola | focus:  | conic section directrix is the Euler line |
| Lemoine inellipse | center:  | foci at the triangle centroid and symmedian point |
| Macbeath inconic | center: nine-point center  | foci at the circumcenter and orthocenter |
| Mandart inellipse | center: mittenpunkt  | contact points form the extouch triangle |
| orthic inconic | center: symmedian point  | contact points form the orthic triangle |
| Steiner inellipse | center: triangle centroid  | tangent at the midpoints of the sides |
| Yff parabola | focus:  | conic section directrix joins the orthocenter and mittenpunkt |
