Isotomic Conjugate

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 product of isotomic and isogonal 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 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 (Casey 1893, Vandeghen 1965). The isotomic conjugate of the line at infinity having trilinear equation

(3)

is the Steiner circumellipse

(4)

(Vandeghen 1965).

The isotomic conjugate of the Euler line is called the Jerabek hyperbola (Casey 1893, Vandeghen 1965).

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 tangent to , the transform is a parabola.

3. If cuts , the transform is a hyperbola, which is a rectangular hyperbola if the line passes through the isotomic conjugate of the orthocenter

(Casey 1893, Vandeghen 1965).