Isotomic Conjugate Point
The point of concurrence 
of the Isotomic Lines relative to a point 
. The isotomic conjugate
of a point with Trilinear Coordinates 
is
 | (1) |
having trilinear equation | (2) |
(Casey 1893, Vandeghen 1965). The isotomic conjugateof the Line at Infinity having trilinear equation | (3) |
 | (4) |
is transformed is determined by whether the line meetsSteiner's Ellipse 
.- 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 passesthrough the isotomic conjugate of the Orthocenter
There are four points which are isotomically self-conjugate: the Centroid 
andeach of the points of intersection of lines through the Vertices Parallel to theopposite sides. The isotomic conjugate of the Euler Line is called Jerabek's Hyperbola (Casey 1893,Vandeghen 1965).
Vandeghen (1965) calls the transformation taking points to their isotomic conjugate points the Cevian Transform. Theproduct of isotomic and Isogonal is a Collineation which transforms the sides of aTriangle to themselves (Vandeghen 1965).
See also Cevian Transform, Gergonne Point, Isogonal Conjugate, Jerabek's Hyperbola, Nagel Point,Steiner's Ellipse
References
Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893. Eddy, R. H. and Fritsch, R. ``The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle.'' Math. Mag. 67, 188-205, 1994. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 157-159, 1929. Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.
- Mill
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