Isogonality

Isogonality is a geometric relation describing symmetry with respect to angle bisectors. In its simplest form, two lines that emanate from the same angle are isogonal if they are mirror images of each other across the angle’s bisector. This idea extends to figures and transformations, where the notion of isogonality captures a balance or interchangeability of directions about a vertex.

In triangle geometry, isogonal conjugation is a central construction. For a triangle ABC and a point P in the plane, reflect the cevians AP, BP, CP about the angle bisectors at A, B, and C, respectively. The three reflected lines concur at a single point P*, called the isogonal conjugate of P with respect to triangle ABC. The isogonal map is an involution: applying it twice returns the original point. This correspondence explains and relates many classical triangle centers. For example, the isogonal conjugate of the centroid is the symmedian point, the isogonal conjugate of the orthocenter is the circumcenter, and the isogonal conjugate of the incenter is the incenter itself.

In three-dimensional geometry, isogonality often refers to vertex transitivity. A polyhedron (or more generally a polytope) is isogonal if its symmetry group acts transitively on its vertices, meaning all vertices are equivalent under symmetries. Isogonal polyhedra are exactly the vertex-transitive solids, a class that includes the regular solids and many Archimedean solids. This usage highlights a uniformity of vertex environment under the figure’s full symmetry group.

The term comes from the Greek for equal angles and isogonality appears in both plane and spatial geometry as a way of expressing symmetric or balanced directional relationships.