On a theorem of Jacob Steiner (Q2372143)

Let \(\mathcal E\) be the real Euclidean \(3\)-space. A tetrahedron \(T\) is called admissible, if opposite edges of \(T\) are not orthogonal. In \(\mathcal E\) Steiner's theorem holds: The four vertex-altitudes of an admissible tetrahedron belong to a regulus \(\mathcal R\). (Reviewer's additions: The four orthocentric perpendiculars are members of the complementary regulus \({\mathcal R}^{c}\) of \(\mathcal R\). For a modern treatment of Steiner's theorem see \textit{H. Havlicek} and \textit{G. Weiß} [Am. Math. Mon. 110, No. 8, 679--693 (2003; Zbl 1048.51009)]. For the three edge-altitudes cf. \textit{G. Weiß} and \textit{H. Havlicek} [KoG 6, 71--80 (2002; Zbl 1086.51506)].) The author investigates the validity of Steiner's theorem in the following two spaces with orthogonality: 1. In an affine \(3\)-space \(\mathcal A\) whose orthogonality is defined, in essential, by a polarity \(\pi\) in the plane at infinity. 2. In a projective \(3\)-space \(\mathcal P\) with given polarity \(\pi\). In the affine case the author exhibits \(8\), in the projective case \(7\) statements that are equivalent with Steiner's theorem. We give a selection; in both cases Steiner's theorem is equivalent to the subsequent two statements: 1. The theorem of Pappus holds and \(\pi\) is described by a symmetric bilinear form. 2. If in \(\mathcal A\) resp. \(\mathcal P\) two pairs of opposite edges of a tetrahedron are orthogonal resp. conjugate, then also the third pair of edges is orthogonal resp. conjugate (``affine resp. projective tetrahedron condition''). In the affine case Steiner's theorem is equivalent to the validity of the ``Joachimsthal condition'' which says: If \(h\) is a vertex-altitude of an admissible tetrahedron, then there exists a parallel to \(h\) which meets all four vertex-altitudes.