Orthologic Tetrahedra with Intersecting Edges

TL;DR

This paper studies a special class of tetrahedra called orthosecting tetrahedra, showing they have six intersection points on a sphere and exploring their geometric properties and conjugate pairs.

Contribution

It introduces the concept of orthosecting tetrahedra, proves their intersection points lie on a sphere, and describes their geometric construction and conjugate relationships.

Findings

Six intersection points lie on a sphere.

Existence of a one-parametric family of orthosecting tetrahedra.

Construction method for conjugate orthosecting tetrahedra.

Abstract

Two tetrahedra are called orthologic if the lines through vertices of one and perpendicular to corresponding faces of the other are intersecting. This is equivalent to the orthogonality of non-corresponding edges. We prove that the additional assumption of intersecting non-corresponding edges ("orthosecting tetrahedra") implies that the six intersection points lie on a sphere. To a given tetrahedron there exists generally a one-parametric family of orthosecting tetrahedra. The orthographic projection of the locus of one vertex onto the corresponding face plane of the given tetrahedron is a curve which remains fixed under isogonal conjugation. This allows the construction of pairs of conjugate orthosecting tetrahedra to a given tetrahedron.