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 $BC, CA, AB$ 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

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.)

Generalization

In may 2021, Dao Thanh Oai given a generalization of Isotomic conjugate as follows:^[1]

Let $△ ABC$ be a triangle, $P$ be a point on its plane and $Ω$ an arbitrary circumconic of $△ ABC$ . Lines $AP, BP, CP$ cuts again $Ω$ at $A', B', C'$ respectively, and parallel lines through these points to $BC, CA, AB$ cut $Ω$ again at $A", B", C"$ respectively. Then lines $AA", BB", CC"$ concurent.

If barycentric coordinates of the center $X$ of $Ω$ are $X=x:y:z:$ and $P=p:q:r$ , then $D$ , the point of intersection of $AA", BB", CC"$ is: $D=D(X,P)=x*(x-y-z)*q*r::$

The point $D$ above call the $X$ -Dao conjugate of $P$ , this conjugate is a generalization of all known kinds of conjugaties:^[1]

When $Ω$ is the circumcircle of $ABC$ , Dao conjugate become the isogonal conjugate of $P$ .
When $Ω$ is the Steiner circumellipse of $ABC$ , Dao conjugate become the isotomic conjugate of $P$ .
When $Ω$ is the circumconic of $ABC$ with center $X$ = $X(1249)$ , Dao conjugate become the Polar conjugate of $P$ .

References

Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, Macmillan and Co., 1893, page 57.
Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 157–159, 278

^ ^a ^b César Eliud Lozada, Preamble before X(44687)Encyclopedia of Triangle Centers

External links

Weisstein, Eric W. "Isotomic Conjugate". MathWorld.
Paul Yiu: Isotomic and isogonal conjugates
Navneel Singhal: Isotomic and isogonal conjugates

[Lozada-1] César Eliud Lozada, Preamble before X(44687)Encyclopedia of Triangle Centers

[1]