Isogonal conjugation in isosceles tetrahedron
Saro Harutyunyan
TL;DR
This work extends the concept of isogonal conjugation from polygons to the isosceles tetrahedron, where opposite edges are equal and the circumcenter coincides with the incenter. It establishes a dual geometric picture: (i) isogonal conjugates symmetric about any bimedian lie on a common hyperbolic paraboloid, and (ii) the circumsphere of an isosceles tetrahedron is invariant under isogonal conjugation (excluding vertices). The authors provide a rigorous coordinate-bashing derivation of the paraboloid loci, prove a pedal-sphere characterization of isogonality, and give inversion-based arguments for circumsphere invariance, complemented by an elegant alternative construction due to a Bogdanov. Together, these results reveal deep symmetry between isogonal conjugation and the circumscribed geometry of isosceles tetrahedra, with contrasts to the planar triangle case.
Abstract
In this article we investigate the properties of isogonal conjugation in isosceles tetrahedron. Particularly we reveal three hyperbolic paraboloids each of which is formed by pairs of isogonal conjugate points symmetric in the respective bimedian, as well as we prove that the circumsphere of an isosceles tetrahedron is invariant under isogonal conjugation in that tetrahedron.