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.

Angle bisectors (concur at incenter $I$ )

  Lines from each vertex to $P$

  Lines to $P$ reflected about the angle bisectors (concur at $P*$ , the **isogonal conjugate** of $P$ )

Isogonal conjugate transformation over the points inside the triangle.

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

A second definition of isogonal conjugate

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]

Properties

The isogonal conjugate of the incenter of triangle $△ ABC$ is the incenter itself.

The isogonal conjugate of the symmedian point is the third Centroid, and the isogonal conjugate of the orthocenter is the circumcenter.

Isogonal conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. This property holds for Isotomic conjugate as well.

Generalization

X-Dao conjugate of P

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 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

^ 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

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