isogonal conjugate

{{Short description|Geometric transformation applied to points with respect to a given triangle}}

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[[Image:Isogonal_Conjugate.svg|right|thumb|

{{legend-line|solid lime|Angle bisectors (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.

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In geometry, the isogonal conjugate of a point {{mvar|P}} with respect to a triangle {{math|△ABC}} is constructed by reflecting the lines {{mvar|PA, PB, PC}} about the angle bisectors of {{mvar|A, B, C}} respectively. These three reflected lines concur at the isogonal conjugate of {{mvar|P}}. (This definition applies only to points not on a 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 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 {{math|△ABC}}, then its isogonal conjugate is $\backslash tfrac\{1\}\{x\}\; :\; \backslash tfrac\{1\}\{y\}\; :\; \backslash tfrac\{1\}\{z\}.$ For this reason, the isogonal conjugate of {{mvar|X}} is sometimes denoted by {{math|X{{sup| –1}}}}. The set {{mvar|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 {{mvar|X}} in {{mvar|S}} is {{math|X{{sup| –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 {{mvar|X}} is on the cubic, then {{math|X{{sup| –1}}}} is also on the cubic.