The isogonal conjugate X^-1 of a point X in the plane of the triangle ΔABC is constructed by reflecting the lines AX, BX, and CX about the angle bisectors at A, B, and C, respectively. The three reflected lines then concur at the isogonal conjugate X^^-1 of X.

The trilinear coordinates of the isogonal conjugate of the point with trilinear coordinates α : β : γ are

α^-1 : β^-1 : γ^-1.

The barycentric coordinates of the isogonal conjugate of the point with barycentric coordinates x : y : z are

a²/x : b²/y : c²/z.

One famous pair of isogonal conjugates is the orthocenter and the circumcenter. The isogonal conjugate of the incenter is itself. More generally, a point X is its own isogonal conjugate with respect to triangle ΔABC if, and only if, X is the incenter or one of the three excenters of triangle ΔABC. The isogonal conjugate of a point on the circumcircle is a point at infinity and vice-versa.

The isogonal conjugate mapping transforms lines into circumconics. The type of conic section is determined by whether the line l meets the circumcircle K. If l does not intersect K, the isogonal image is an ellipse, if l is tangent to K, the isogonal image is a parabola and if l cuts K in two points, the isogonal image is a hyperbola, which is a rectangular hyperbola if the line passes through the circumcenter.

In the applet below, P and Q are isogonal conjugates. The following relationship holds: s/r = x/y, where s = d(P,S), r = d(P,R), x = d(Q,X) and y = d(Q,Y). Also, the angles indicate in the applet are congruent.

References:
Darij Grinberg, Isogonal Conjugation with Respect to a Triangle (version 23 September 2006).
Ross Honsbeger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry. The Mathematical Association of America, 1996.
Eric W. Weisstein, Isogonal Conjugate. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IsogonalConjugate.html.