In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the isotomic conjugate of a point P not on a sideline of triangle ABC is constructed as follows: Let A', B', C' be the points in which the lines AP, BP, CP meet the lines BC, CA, AB, respectively. Reflect A'B'C' in the midpoints of sides BC, CA, AB to obtain points A", B", C", respectively. The lines AA", BB", CC" meet at a point (this can be proved using Ceva's theorem

Ceva's theorem

Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively...

), and this point is called the isotomic conjugate of P.

If trilinears

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...

 for P are p : q : r, then trilinears for the isotomic conjugate of P are

a⁻²p⁻¹ : b⁻²q⁻¹ : c⁻²r⁻¹.

The isotomic conjugate of the centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 of triangle ABC is the centroid itself.

Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.)