We can infer the proof from existence of isogonal conjugate too.

The isogonal conjugate of the incentre I is itself.

The Brocard points are isogonal conjugates of each other.

The isogonal conjugate of the circumcircle is the line at infinity.

The isogonal conjugate and also the complement of the orthocenter is the circumcenter.

The isogonal conjugate of a triangle's centroid is its symmedian point.

(This property holds for isogonal conjugates as well.)

Isogonal conjugate in triangle geometry.

The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Y of X.

The isodynamic points are the isogonal conjugates of the two Fermat points of triangle , and vice versa.

The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates.

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 each point P on the cubic but not on a sideline of the cubic, the isogonal conjugate of P is also on the cubic.

The isogonal conjugate of the centroid X is the symmedian point X (also denoted by K) having trilinear coordinates ( a : b : c ).

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 nine-point center of triangle ABC is the Kosnita point X of triangle ABC.

For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, called the isogonal conjugate of P.

The Darboux cubic may be defined from the de Longchamps point, as the locus of points such that , the isogonal conjugate of , and the de Longchamps point are collinear.

The isogonal conjugate of the circumcenter X is the orthocenter X (also denoted by H) having trilinear coordinates ( sec A : sec B : sec C ).

The isogonal conjugate of the orthocentre H is the circumcentre O. The isogonal conjugate of the centroid G is (by definition) the symmedian point K. The isogonal conjugates of the Fermat points are the isodynamic points and vice versa.