# Isogonal conjugate

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Geometric transformation applied to points with respect to a given triangle

  [Angle bisectors](/source/Angle_bisector) ([concur](/source/Concurrent_lines) at [incenter](/source/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.

In [geometry](/source/Geometry), the **isogonal conjugate** of a [point](/source/Point_(geometry)) P with respect to a [triangle](/source/Triangle) △*ABC* is constructed by [reflecting](/source/Reflection_(mathematics)) the lines PA, PB, PC about the [angle bisectors](/source/Angle_bisectors) of A, B, C respectively. These three reflected lines [concur](/source/Concurrent_lines) at the isogonal conjugate of P. (This definition applies only to points not on a [sideline](/source/Extended_side) of triangle △*ABC*.) This is a direct result of the trigonometric form of [Ceva's theorem](/source/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](/source/Incentre) I is itself. The isogonal conjugate of the [orthocentre](/source/Orthocentre) H is the [circumcentre](/source/Circumcentre) O. The isogonal conjugate of the [centroid](/source/Centroid) G is (by definition) the [symmedian point](/source/Symmedian_point) K. The isogonal conjugates of the [Fermat points](/source/Fermat_point) are the [isodynamic points](/source/Isodynamic_point) and vice versa. The [Brocard points](/source/Brocard_points) are isogonal conjugates of each other.

In [trilinear coordinates](/source/Trilinear_coordinates), if X = x : y : z {\displaystyle X=x:y:z} is a point not on a sideline of triangle △*ABC*, then its isogonal conjugate is 1 x : 1 y : 1 z . {\displaystyle {\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](/source/Set_(mathematics)) S of triangle centers under the trilinear product, defined by

- ( p : q : r ) ∗ ( u : v : w ) = p u : q v : r w , {\displaystyle (p:q:r)*(u:v:w)=pu:qv:rw,}

is a [commutative group](/source/Commutative_group), and the inverse of each X in S is *X* –1.

As isogonal conjugation is a [function](/source/Function_(mathematics)), 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](/source/Circumconic_and_inconic); specifically, an [ellipse](/source/Ellipse), [parabola](/source/Parabola), or [hyperbola](/source/Hyperbola) according as the line intersects the [circumcircle](/source/Circumcircle) in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the [line at infinity](/source/Line_at_infinity). Several well-known [cubics](/source/Cubic_plane_curve) (e.g., [Thompson cubic](/source/Thomson_cubic), Darboux cubic, [Neuberg cubic](/source/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 Pa, Pb, Pc. Then the center of the circle 〇*PaPbPc* is the isogonal conjugate of P.[1]

## See also

- [Isotomic conjugate](/source/Isotomic_conjugate)

- [Central line (geometry)](/source/Central_line_(geometry))

- [Triangle center](/source/Triangle_center)

## References

1. **[^](#cite_ref-1)** Steve Phelps. ["Constructing Isogonal Conjugates"](https://www.geogebra.org/m/sRVERPyd). *GeoGebra*. GeoGebra Team. Retrieved 17 January 2022.

## External links

Wikimedia Commons has media related to [Isogonal Conjugates](https://commons.wikimedia.org/wiki/Category:Isogonal_Conjugates).

- [Interactive Java Applet illustrating isogonal conjugate and its properties](https://web.archive.org/web/20110319132218/http://www.uff.br/trianglecenters/isogonal-conjugate_en.html)

- [MathWorld](http://mathworld.wolfram.com/IsogonalConjugate.html)

- [Pedal Triangle and Isogonal Conjugacy](http://www.cut-the-knot.org/Curriculum/Geometry/OrthologicPedal.shtml)

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Adapted from the Wikipedia article [Isogonal conjugate](https://en.wikipedia.org/wiki/Isogonal_conjugate) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Isogonal_conjugate?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.