
Journal for Geometry and Graphics 18 (2014), No. 2, 205215 Copyright Heldermann Verlag 2014 On a 3D Extension of the SimsonWallace Theorem Pavel Pech Faculty of Education, University of South Bohemia, Jeronýmova 10, 371 15 Ceské Budejovice, Czech Republic pech@pf.jcu.cz The following 3D extension of the SimsonWallace theorem is proved by a method which differs from that used in the past (Theorem 1): Let K, L, M, N be orthogonal projections of a point P to the faces BCD, ACD, ABD, and ABC of a tetrahedron ABCD. Then, all points P with the property that the tetrahedron KLMN has a constant volume belong to a cubic surface. Next, the main theorem (Theorem 2) is proved which states that also the converse of Theorem 1 holds. Furthermore, we verify Theorem 2 for a regular tetrahedron by descriptive geometry methods using dynamic geometry software. To do this we take advantage of the fact that this cubic surface can be represented by a parametric system of conics which lie in mutually parallel planes. Keywords: SimsonWallace loci, cubic surface, Cayley cubic, Monge projection. MSC: 51N20; 51N05, 51N35 [ Fulltextpdf (460 KB)] for subscribers only. 