Given a line having trilinear coordinate equation
 |
with respect to a reference triangle 
, the point
 |
is called the trilinear pole of the line with respect to 
(Kimberling 1998, p. 38).
The following table gives the trilinear poles for some named lines.
| line |  | trilinear pole | Kimberling center |
| antiorthic axis |  | incenter  |  |
| Brocard axis |  | focus of the Kiepert parabola |  |
| de Longchamps line |  | third
Brocard point  |  |
| Euler line |  |  | |
| Fermat axis |  | Tixier point |  |
| Gergonne line |  | Gergonne
point  |  |
| Lemoine axis |  | symmedian
point  |  |
| line at infinity |  | triangle centroid  |  |
| Nagel line |  | Yff parabolic point |  |
| orthic axis |  | orthocenter  |  |
| Soddy line |  |  | |
| van Aubel line |  |  |
Let 
be the intercept of the line 
and 
, and let 
be the harmonic conjugate
of 
with respect to 
and 
.
Similarly define 
and 
.
Then 
is the Cevian triangle of the trilinear pole of
.
