Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space

With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points. 1. The Problem in Perspective Perhaps the most important proposition deduced from the axioms of incidence in projective geometry for the projective space is Desargues’ two-triangle Theorem usually stated quite concisely as follows: two triangles in space are perspective from a point if and only if they are perspective from a line meaning that if we assume a one-to-one correspondence among the vertices of the two triangles, then the lines joining the corresponding vertices are concurrent if and only if the intersections of the lines of the corresponding sides are collinear (Figure 1). According to the ancient Greek mathematician Pappus (3rd century A.D.), this theorem was essentially contained in the lost treatise on Porisms of Euclid (3rd century B.C.) [1, 2] but it is nowadays known by the name of the French mathematician and military engineer Gerard Desargues (1593–1662) who published it in 1639. Actually only half of it is called Desargues’ Theorem (perspectivity from a point implies perspectivity from a line) whereas the other half is called converse of Desargues’ Theorem. All printed copies of Desargues’ treatise were lost, but fortunately Desargues’ contemporary French mathematician Phillipe de La Hire (1640–1718) made a manuscript copy of it which was discovered again some 200 years later [3]. Figure 1: Desargues’ 2-triangle Theorem in space and its converse. There exist a few hidden assumptions in the above pretty and compact statement of the theorem. Namely, all mentioned lines and intersection points are assumed to exist and the three points on each plane are assumed to be noncollinear. It seems that in bibliography there exist very few treatments of the theorem paying attention to these assumptions, like Hodge and Pedoe’s [4] and Pogorelov’s [5]. The purpose of this paper is to deal with these assumptions. It is also worth mentioning that Desargues was interested in triangles in space formed by intersecting two distinct planes with three lines going through a common point. Remarkably, although the theorem remains true even when the two planes of the projective space coincide, it is not at all