Isogonal conjugate

File:Isogonal Conjugate.svg

  Angle bisectors (concur at incenter Template:Mvar)

  Lines from each vertex to Template:Mvar

  Lines to Template:Mvar reflected about the angle bisectors (concur at Template:Mvar, the isogonal conjugate of Template:Mvar)

File:Isogonal Conjugate transform.svg

Isogonal conjugate transformation over the points inside the triangle.

In geometry, the isogonal conjugate of a point Template:Mvar with respect to a triangle △ABC is constructed by reflecting the lines Template:Mvar about the angle bisectors of Template:Mvar respectively. These three reflected lines concur at the isogonal conjugate of Template:Mvar. (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 Template:Mvar is sometimes denoted by Template:Mvar. The isogonal conjugate of Template:Mvar is Template:Mvar.

The isogonal conjugate of the incentre Template:Mvar is itself. The isogonal conjugate of the orthocentre Template:Mvar is the circumcentre Template:Mvar. The isogonal conjugate of the centroid Template:Mvar is (by definition) the symmedian point Template:Mvar. 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 $\frac{1}{x}:\frac{1}{y}:\frac{1}{z}.$ For this reason, the isogonal conjugate of Template:Mvar is sometimes denoted by X ^–1. The set Template:Mvar 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 Template:Mvar in Template:Mvar 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 Template:Mvar is on the cubic, then X ^–1 is also on the cubic.

Another construction for the isogonal conjugate of a point

File:A Second Definition Of Isogonal Conjugate.png

A second definition of isogonal conjugate

For a given point Template:Mvar in the plane of triangle △ABC, let the reflections of Template:Mvar in the sidelines Template:Mvar be Template:Mvar. Then the center of the circle 〇P_aP_bP_c is the isogonal conjugate of Template:Mvar.

See also

• Isotomic conjugate
• Central line (geometry)
• Triangle center

External links

• Interactive Java Applet illustrating isogonal conjugate and its properties
• MathWorld
• Pedal Triangle and Isogonal Conjugacy