  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.
