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]

References

^ Steve Phelps. "Constructing Isogonal Conjugates". GeoGebra. GeoGebra Team. Retrieved 17 January 2022.

External links

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

[1]

Another construction for the isogonal conjugate of a point

See also

References

External links