{{Short description|Geometric transformation applied to points with respect to a given triangle}} __NOTOC__ [[Image:Isogonal_Conjugate.svg|right|thumb| {{legend-line|solid lime|[[Angle bisector]]s ([[Concurrent lines|concur]] at [[incenter]] {{mvar|I}})}} {{legend-line|solid blue|Lines from each vertex to {{mvar|P}}}} {{legend-line|solid red|Lines to {{mvar|P}} reflected about the angle bisectors (concur at {{mvar|P*}}, the '''isogonal conjugate''' of {{mvar|P}})}}]] [[Image:Isogonal_Conjugate_transform.svg|right|thumb| Isogonal conjugate transformation over the points inside the triangle. ]]

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

The isogonal conjugate of a point {{mvar|P}} is sometimes denoted by {{mvar|P*}}. The isogonal conjugate of {{mvar|P*}} is {{mvar|P}}.

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

In [[trilinear coordinates]], if <math>X=x:y:z</math> is a point not on a sideline of triangle {{math|△''ABC''}}, then its isogonal conjugate is <math>\tfrac{1}{x} : \tfrac{1}{y} : \tfrac{1}{z}.</math> For this reason, the isogonal conjugate of {{mvar|X}} is sometimes denoted by {{math|''X''{{sup| –1}}}}. The [[Set (mathematics)|set]] {{mvar|S}} of triangle centers under the trilinear product, defined by

: <math>(p:q:r)*(u:v:w) = pu:qv:rw,</math>

is a [[commutative group]], and the inverse of each {{mvar|X}} in {{mvar|S}} is {{math|''X''{{sup| –1}}}}.

As isogonal conjugation is a [[Function (mathematics)|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 and inconic|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 [[Cubic plane curve|cubics]] (e.g., [[Thomson cubic|Thompson cubic]], Darboux cubic, [[Neuberg cubic]]) are self-isogonal-conjugate, in the sense that if {{mvar|X}} is on the cubic, then {{math|''X''{{sup| –1}}}} is also on the cubic.

==Another construction for the isogonal conjugate of a point== [[File:A Second Definition Of Isogonal Conjugate.png|thumb|A second definition of isogonal conjugate]] For a given point {{mvar|P}} in the plane of triangle {{math|△''ABC''}}, let the reflections of {{mvar|P}} in the sidelines {{mvar|BC, CA, AB}} be {{mvar|P{{sub|a}}, P{{sub|b}}, P{{sub|c}}}}. Then the center of the circle {{math|〇''P{{sub|a}}P{{sub|b}}P{{sub|c}}''}} is the isogonal conjugate of {{mvar|P}}.<ref>{{cite web |last1=Steve Phelps |title=Constructing Isogonal Conjugates |url=https://www.geogebra.org/m/sRVERPyd |website=GeoGebra |publisher=GeoGebra Team |access-date=17 January 2022}}</ref>

== See also == * [[Isotomic conjugate]] * [[Central line (geometry)]] * [[Triangle center]]

==References== {{reflist}}

==External links== {{commons category|Isogonal Conjugates}} * [https://web.archive.org/web/20110319132218/http://www.uff.br/trianglecenters/isogonal-conjugate_en.html Interactive Java Applet illustrating isogonal conjugate and its properties] *[http://mathworld.wolfram.com/IsogonalConjugate.html MathWorld] * [http://www.cut-the-knot.org/Curriculum/Geometry/OrthologicPedal.shtml Pedal Triangle and Isogonal Conjugacy]

[[Category:Triangle geometry]]