Geometry

Volume 2014 (2014), Article ID 276108, 7 pages

http://dx.doi.org/10.1155/2014/276108

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

Department of Applied Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 22 August 2014; Accepted 15 November 2014; Published 18 December 2014

Academic Editor: Isaac Pesenson

Copyright © 2014 Dimitrios Kodokostas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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].