Isotomic conjugate

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 $△ABC$: If the base points of the lines $PA, PB, PC$ on the sides opposite $A, B, C$ are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of $P$.

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 $\overline{BC}, \overline{CA}, \overline{AB}$ 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


 * $$a^{-2}p^{-1} : b^{-2}q^{-1} : c^{-2}r^{-1},$$

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.)