Isogonal conjugate
In geometry, the isogonal conjugate of a point Template:Mvar with respect to a triangle Template:Math 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 Template:Math.) 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 is a point not on a sideline of triangle Template:Math, then its isogonal conjugate is For this reason, the isogonal conjugate of Template:Mvar is sometimes denoted by Template:Math. The set Template:Mvar of triangle centers under the trilinear product, defined by
is a commutative group, and the inverse of each Template:Mvar in Template:Mvar is Template:Math.
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 Template:Math is also on the cubic.
Another construction for the isogonal conjugate of a point
For a given point Template:Mvar in the plane of triangle Template:Math, let the reflections of Template:Mvar in the sidelines Template:Mvar be Template:Mvar. Then the center of the circle Template:Math is the isogonal conjugate of Template:Mvar.[1]