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Journal for Geometry and Graphics 18 (2014), No. 2, 205--215
Copyright Heldermann Verlag 2014

On a 3D Extension of the Simson-Wallace Theorem

Pavel Pech
Faculty of Education, University of South Bohemia, Jeronýmova 10, 371 15 Ceské Budejovice, Czech Republic

The following 3D extension of the Simson-Wallace 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: Simson-Wallace loci, cubic surface, Cayley cubic, Monge projection.

MSC: 51N20; 51N05, 51N35

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