A Characterization by Optimization of the Monge Point of a Tetrahedron

“... nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluceat”, translated into “... nothing in all the world will occur in which no maximum or minimum rule is somehow shining forth”, used to say L.Euler in 1744. This is confirmed by numerous applications of mathematics in physics, mechanics, economy, etc. In this note, we show that it is also the case for the classical “centers” of a tetrahedron, more specifically for the so-called Monge point (the substitute of the notion of orthocenter for a tetrahedron). To the best of our knowledge, the characterization of the Monge point of a tetrahedron by optimization, that we are going to present, is new. 1. To begin with... What kind of tetrahedron? Let T = ABCD be a tetrahedron in the three dimensional space R(equipped with the usual Euclidean and affine structures); the points A,B,C,D are supposed not to lie in a plane, of course. We begin with two particular types of tetrahedra and, then, with increase in generality, we can classify the tetrahedra into several classes. Here they are: The regular tetrahedron. This tetrahedron enjoys so many symmetries that it is not very interesting from the optimization viewpoint: all the “centers” usually associated with a tetrahedron (and that we are going to visit again in the next paragraph) coincide. The trirectangular tetrahedra. They are generalizations to the space of rectangular triangles in the plane. A trirectangular tetrahedron OABC has (two by two) three perpendicular faces OBC,OAB,OAC and a “hypothenuse-face” ABC; such a tetrahedron enjoys a remarkable relationship between areas of its faces (see [1]); its vertex O, opposite the hypothenuse-face, is the orthocenter and Monge point, as we shall see below. The orthocentric tetrahedra. Curiously enough, the four altitudes of a tetrahedron generally do not meet at a point; when this happens, the tetrahedron is called orthocentric. A common characterization of orthocentric tetrahedra is as follows: a tetrahedron is orthocentric if and only if the opposite edges (two by two) are orthogonal. This class of tetrahedra is by far the most studied one in the literature. Regular and trirectangular tetrahedra are indeed orthocentric. General tetrahedra. Like for triangles, three specific “centers” can be defined for any tetrahedron: the centroid or isobarycenter, the incenter and the circumcenter. We shall see their characterization by optimization, as for some other points, in the next section. As said before, the altitudes do not necessarily meet at a point; moreover, the projection of any vertex on the opposite face does not necessarily coincide with the orthocenter of this face. The notion of orthocenter will be held by a new point: the so-called Monge point.