Isogonal Conjugate

The isogonal conjugate of a point in the plane of the triangle is constructed by reflecting the lines , , and about the angle bisectors at , , and . The three reflected lines then concur at the isogonal conjugate (Honsberger 1995, pp. 55-56). In older literature, isogonal conjugate points are also known as counter points (Gallatly 1913), Gegenpunkte (Gallatly 1913), and focal pairs (Morley 1954).

The trilinear coordinates of the isogonal conjugate of the point with coordinates

(1)

are

(2)

The following labels lists a number of isogonal conjugate pairs of points.

triangle center	isogonal conjugate
circumcenter	orthocenter
Clawson point
de Longchamps point
Feuerbach point
first Brocard point	second Brocard point
first Fermat point	first isodynamic point
first isodynamic point	first Fermat point
Gergonne point
incenter	incenter
second Napoleon point
isogonal mittenpunkt	mittenpunkt
Kosnita point Ko	nine-point center
mittenpunkt	isogonal mittenpunkt
Nagel point Na
nine-point center	Kosnita Point Ko
orthocenter	circumcenter
orthocenter of the orthic triangle
first Napoleon point
Schiffler point	orthocenter of the contact triangle
second Brocard point	first Brocard point
second Fermat point	second isodynamic point
second isodynamic point	second Fermat point
Spieker center Sp
symmedian point	triangle centroid
symmedian point of the orthic triangle
third Brocard point	third power point
triangle centroid	symmedian point

In the above figure with and isogonal conjugates,

(3)

(Honsberger 1995, pp. 54-55).

Isogonal conjugation maps the interior of a triangle onto itself. This mapping transforms lines into circumconics. The type of conic section is determined by whether the line meets the circumcircle ,

1. If does not intersect , the isogonal transform is an ellipse;

2. If is tangent to , the transform is a parabola;

3. If cuts , the transform is a hyperbola, which is a rectangular hyperbola if the line passes through the circumcenter

(Casey 1893, Vandeghen 1965).

The isogonal conjugate of a point on the circumcircle is a point at infinity (and conversely). The sides of the pedal triangle of a point are perpendicular to the connectors of the corresponding polygon vertices with the isogonal conjugate. The isogonal conjugate of a set of points is the locus of their isogonal conjugate points.

The product of isotomic and isogonal conjugation is a collineation which transforms the sides of a triangle to themselves (Vandeghen 1965).

A spatial analog is defined for an isosceles tetrahedron by taking reciprocal barycentric coordinates. The four face areas are equal, and the conjugate is finite when the sum of the four reciprocals is nonzero (Harutyunyan 2025).