Geodesic
On a 3D Extension of the Simson-Wallace Theorem
Journal for geometry and graphics, Tome 18 (2014) no. 2, pp. 205-215 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 51N20, 51N05, 51N35
Mots-clés : Simson-Wallace loci, cubic surface, Cayley cubic, Monge projection 
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     author = {P. Pech },
     title = {On a {3D} {Extension} of the {Simson-Wallace} {Theorem}},
     journal = {Journal for geometry and graphics},
     pages = {205--215},
     year = {2014},
     volume = {18},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JGG_2014_18_2_JGG_2014_18_2_a5/}
}
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P. Pech . On a 3D Extension of the Simson-Wallace Theorem. Journal for geometry and graphics, Tome 18 (2014) no. 2, pp. 205-215. http://geodesic.mathdoc.fr/item/JGG_2014_18_2_JGG_2014_18_2_a5/