Orthologic tetrahedra with intersecting edges. (English) Zbl 1192.51012

Schröcker, Hans-Peter

KoG 13, 13-18 (2009).

Summary: 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.

MSC:

51M04

Elementary problems in Euclidean geometries

Keywords:

orthologic tetrahedra; orthosecting tetrahedra; isogonal conjugate