Isogonal conjugate

In geometry, the isogonal conjugate of a point $P$ with respect to a triangle $△ ABC$ is constructed by reflecting the lines $PA, PB, PC$ about the angle bisectors of $A, B, C$ respectively. These three reflected lines concur at the isogonal conjugate of $P$ . (This definition applies only to points not on a sideline of triangle $△ ABC$ .) This is a direct result of the trigonometric form of Ceva's theorem.

The isogonal conjugate of a point $P$ is sometimes denoted by $P*$ . The isogonal conjugate of $P*$ is $P$ .

The isogonal conjugate of the incentre $I$ is itself. The isogonal conjugate of the orthocentre $H$ is the circumcentre $O$ . The isogonal conjugate of the centroid $G$ is (by definition) the symmedian point $K$ . The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points are isogonal conjugates of each other.

In trilinear coordinates, if $X=x:y:z$ is a point not on a sideline of triangle $△ ABC$ , then its isogonal conjugate is ${\tfrac {1}{x}}:{\tfrac {1}{y}}:{\tfrac {1}{z}}.$ For this reason, the isogonal conjugate of $X$ is sometimes denoted by $X -1$ . The set $S$ of triangle centers under the trilinear product, defined by

(p:q:r)*(u:v:w)=pu:qv:rw,

is a commutative group, and the inverse of each $X$ in $S$ is $X -1$ .

As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the line at infinity. Several well-known cubics (e.g., Thompson cubic, Darboux cubic, Neuberg cubic) are self-isogonal-conjugate, in the sense that if $X$ is on the cubic, then $X -1$ is also on the cubic.

Another construction for the isogonal conjugate of a point

For a given point $P$ in the plane of triangle $△ ABC$ , let the reflections of $P$ in the sidelines $BC, CA, AB$ be $P a, P b, P c$ . Then the center of the circle $〇 P a P b P c$ is the isogonal conjugate of $P$ .^[1]

Generalization

In may 2021, Dao Thanh Oai given a generalization of isogonal conjugate as follows:^[2] Let $△ ABC$ be a triangle, $P$ a point on its plane and $Ω$ an arbitrary circumconic of $△ ABC$ . Lines $AP, BP, CP$ cut 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:^[2]

When $Ω$ is the circumcircle of $ABC$ , Dao's generalization become the isogonal conjugate of $P$ .
When $Ω$ is the Steiner circumellipse of $ABC$ , Dao's generalization become the isotomic conjugate of $P$ .
When $Ω$ is the circumconic of $ABC$ Dao's generalization become the Polar conjugate of $P$ .

References

^ Steve Phelps. "Constructing Isogonal Conjugates". GeoGebra. GeoGebra Team. Retrieved 17 January 2022.
^ ^a ^b César Eliud Lozada, Preamble before X(44687)Encyclopedia of Triangle Centers

External links

[1] Steve Phelps. "Constructing Isogonal Conjugates". GeoGebra. GeoGebra Team. Retrieved 17 January 2022.

[Lozada-2] César Eliud Lozada, Preamble before X(44687)Encyclopedia of Triangle Centers

[1]

[2]

Another construction for the isogonal conjugate of a point

Generalization

See also

References

External links