In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of .
We assume that is not collinear with any two vertices of . Let be the points in which the lines meet sidelines (extended if necessary). Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of .
If the trilinears for are, then the trilinears for the isotomic conjugate of are
a-2p-1:b-2q-1:c-2r-1,
where are the side lengths opposite vertices respectively.
The isotomic conjugate of the centroid of triangle 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.)
In may 2021, Dao Thanh Oai given a generalization of Isotomic conjugate as follows:[1]
Let be a triangle, be a point on its plane and an arbitrary circumconic of . Lines cuts again at respectively, and parallel lines through these points to cut again at respectively. Then lines concurent.
If barycentric coordinates of the center of are
X=x:y:z:
P=p:q:r
D=D(X,P)=x*(x-y-z)*q*r::
The point above call the -Dao conjugate of, this conjugate is a generalization of all known kinds of conjugaties:[1]