In geometry, the isotomic conjugate of a point P with respect to a triangle △ABC is another point, defined in a specific way from P and △ABC

Construction

We assume that P is not collinear with any two vertices of △ABC . Let A', B', C' be the points in which the lines AP, BP, CP meet sidelines BC, CA, AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B", C" respectively. The isotomic lines AA", BB", CC" joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of P.

Coordinates

If the trilinears for P are p : q : r, then the trilinears for the isotomic conjugate of P are where a, b, c are the side lengths opposite vertices A, B, C respectively.

Properties

The isotomic conjugate of the centroid of triangle △ABC is the centroid itself. The isotomic conjugate of the symmedian point is the third Brocard point, and the isotomic conjugate of the Gergonne point is the Nagel point. Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.)

Generalization

In may 2021, given a generalization of Isotomic conjugate as follows: Let △ ABC be a triangle, P be a point on its plane and Ω an arbitrary of △ ABC . Lines AP, BP, CP cuts again Ω at A', B', C' respectively, and through these to BC, CA, AB cut Ω again at A", B", C" respectively. Then lines AA", BB", CC" . If barycentric coordinates of the center X of Ω are and, then D , the point of intersection of AA", BB", CC" is: The point D above call the X -Dao conjugate of P , this conjugate is a generalization of all known kinds of conjugaties: Ω is the of ABC , Dao conjugate become the isogonal conjugate of P . Ω is the of ABC , Dao conjugate become the isotomic conjugate of P . Ω is the circumconic of ABC with center X

X(1249) , Dao conjugate become the Polar conjugate of P .