Isogonal

Isogonal describes a symmetry of directions with respect to an angle. Specifically, two lines through a common vertex are isogonal if one is the reflection of the other across the angle’s bisector. The concept is most often used in triangle geometry, where it is extended to isogonal conjugation.

In a triangle ABC, a point P in the plane has an isogonal conjugate P' with respect to ABC. This means that the line AP is reflected about the angle A’s bisector to give AP', and similarly BP' is the reflection of BP about B’s bisector, and CP' is the reflection of CP about C’s bisector. The map P -> P' is called isogonal conjugation and is an involution: applying it again returns the original point.

Isogonal conjugation has several classical consequences. The isogonal of a triangle’s median is the corresponding symmedian; thus the isogonal image of the centroid G is the Lemoine (symmedian) point K. The isogonal of the circumcenter O is the orthocenter H, since AO, BO, CO are reflected to the altitudes AH, BH, CH. The incenter I is fixed by the isogonal map, because AI, BI, CI lie along the angle bisectors.

Applications of isogonality appear in Ceva-type arguments, the study of special triangle centers, and the characterization of lines and curves under reflection about angle bisectors. The concept generalizes to higher settings by considering reflections across the angle bisectors at the reference vertices, though its most prominent form remains within triangle geometry.