The isotomic conjugate of a point is the point of concurrence 
of the isotomic lines relative
to a point 
. The isotomic conjugate 
of a point
with trilinear coordinates 
is
 |
(1)
|
Vandeghen (1965) calls the transformation taking points to their isotomic conjugates the Cevian transform. The function composition of isotomic conjugation and isogonal conjugation is a collineation which transforms the sides of a triangle to themselves (Vandeghen 1965).
An isotomic transversal is sometimes referred to as an isotomic conjugate (Ehrmann and van Lamoen 2004).
There are four points which are isotomically self-conjugate: the triangle centroid 
and each of the exmedian points.
The following table lists some common triangle centers
and their isotomic conjugates.
| triangle center | isotomic conjugate |
| Gergonne point Ge | Nagel point Na |
incenter  |  |
| Nagel point Na | Gergonne point Ge |
orthocenter  | symmedian
point of the anticomplementary triangle  |
symmedian point  | third
Brocard point  |
third Brocard point  | symmedian
point  |
triangle centroid  | triangle
centroid  |
The isotomic conjugate of a line 
having trilinear equation
 |
(2)
|
is a conic section circumscribed on the triangle 
, provided 
passes through none of 
, 
, and 
(Casey 1893, Vandeghen 1965). The isotomic conjugate of the
line at infinity having trilinear equation
 |
(3)
|
is the Steiner circumellipse
 |
(4)
|
(Vandeghen 1965).
When nondegenerate, the isotomic conjugate of the Euler line is a circumhyperbola.
The type of conic section to which 
is transformed is determined by whether the line 
meets the Steiner
circumellipse 
.
1. If 
does not intersect 
, the isotomic transform
is an ellipse.
2. If 
is a tangent line to 
, the isotomic transform
is a parabola.
3. If 
cuts 
, the isotomic transform
is a hyperbola, which is a rectangular
hyperbola if the line passes through the isotomic conjugate
of the orthocenter
(Casey 1893, Vandeghen 1965).
