In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (This definition applies only to points not on a sideline of triangle .) This is a direct result of the trigonometric form of Ceva's theorem.
The isogonal conjugate of a point is sometimes denoted by . The isogonal conjugate of is .
The isogonal conjugate of the incentre is itself. The isogonal conjugate of the orthocentre is the circumcentre . The isogonal conjugate of the centroid is (by definition) the symmedian point . 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 is a point not on a sideline of triangle , then its isogonal conjugate is For this reason, the isogonal conjugate of is sometimes denoted by . The set of triangle centers under the trilinear product, defined by
is a commutative group, and the inverse of each in is .
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 is on the cubic, then is also on the cubic.
For a given point in the plane of triangle , let the reflections of in the sidelines be . Then the center of the circle is the isogonal conjugate of .