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

With the use of only the incidence axiomsweprove and generalizeDesargues’ two-triangleTheorem 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.


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].
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 evident that it holds whenever we work solely in a general projective plane disregarding a larger surrounding projective space. Actually Desargues' Theorem does not hold in the projective geometry of a general projective plane defined by the usual incidence axioms as Hilbert has shown [6], unless if the theorem itself or a statement that implies it is assumed as an axiom. Such a set of axioms was given by Bachmann [7], and a proof of Desargues' Theorem can be found in [8] based on the following Pappus' Theorem of Euclidean Geometry being considered as an axiom: if the six vertices of a hexagon lie alternatively on two lines, then the three points of intersection of pairs of opposite sides are collinear.
(3 ) p Desargues' Theorem does or rather does not hold. They are defined as collections of two kinds of objects called points and lines satisfying at least the following three incidence axioms: (i) any two distinct points are incident with exactly one line; (ii) any two distinct lines are incident with exactly one point; (iii) there exist four points of which no three are collinear, where incidence is used as a neutral word meaning that the point belongs or lies on the line and also meaning that the line passes through or contains the point. Collinearity has the usual meaning. Some projective planes have only finitely many points and lines whereas some others have infinitely many. Desargues' Theorem fails to hold for many of them, both finite and infinite [9][10][11][12]. Hilbert [6], Hall [13], and others have proved a plane is Desarguesian (for such planes Desargues' Theorem holds) if and only if it can be constructed algebraically from the vector spaces 3 over the division rings , where the points of these planes are just the lines of the vector space 3 and the lines of the planes are the subspaces of 3 spanned by two linearly independent vectors of it. The usual real projective plane comes by this construction from R 3 .
Nevertheless, for projective spaces of dimension at least three which are defined similar to the projective planes either by a set of incidence axioms or by algebraic constructions, Desargues' Theorem is always true [4]. These higher dimensional projective spaces contain 2-dimensional projective planes and it is easily seen that Desargues' Theorem holds for all such projective planes too.
We are going to prove and generalize Desargues' Theorem working in a three-dimensional projective space and considering an arbitrary number of points on each one of the two distinct planes, allowing some of the lines and intersection points mentioned in the above statement of the theorem to disappear. For this matter we will allow some of the given points on one of the planes to coincide with their corresponding points on the other plane. Also, although we will always be assuming that the given points are distinct on each plane, we will drop the restriction that any three of them are noncollinear. All our proofs will be based directly on the axioms of projective geometry in space and their immediate corollaries.
The projective space we will be working on consists of three kinds of objects called points, lines, and planes for which the following incidence axioms hold [5]: (i) there exists exactly one line through any two distinct points; (ii) any two distinct lines on the same plane have a unique common point; (iii) there exist at least three points on each line; (iv) there exist three points not on the same line; (v) there exist four points not on the same plane; (vi) there exists exactly one plane through any three distinct noncollinear points; there exists at least one point on each plane; (vii) if two distinct points are on a plane, then all points in the line containing the given points are on the same plane; (viii) if two planes have a common point, then they have at least one more common point.
Some immediate corollaries are the following: (i) a plane and a line not on the plane always have a unique common point; (ii) two distinct planes intersect in a common line.
A number of given points will be called collinear whenever some line contains all of them (if some points coincide, there may be more than one such line). Similarly, a number of given lines will be called concurrent whenever there exists at least one point lying on all of them. We will say a given set of points on a plane is in general position whenever no three collinear points exist among them. A usual triangle or just a triangle will be a triad of lines defined by three noncollinear points. Figure 2(a) makes it clear that if we wish to generalize to larger numbers of points the converse of Desargues' Theorem, we have to assume distinct given points on each plane. Similarly, Figure 2(b) says that we should not allow some three collinear points on a plane whenever some of the rest coincide with their corresponding points on the other plane.

A Generalized Desargues' Theorem for Points in General Position Allowing Corresponding Points to Coincide
Nevertheless there do exist generalizations if we allow only one of these two bothersome situations to happen, as we show below. So let (1), (2), . . . be given points on a plane and (1 ), (2 ), . . . some corresponding points on another plane allowing for the possibility that some corresponding points coincide. Instead of ignoring pairs of coinciding corresponding points we choose to use generalized lines (2 ) (1) (3 ) p defining a generalized line ( )( ) to be the line through ( ) and ( ) whenever ( ), ( ) are distinct, otherwise to be the set of all lines through ( ) as this seems most natural. It is also convenient to say that ( )( ) contains or goes through all points in our projective space. Our first generalization is the following (Figure 3). provides a quick pictorial proof) and so they should have a nonempty intersection. But the first of these lines belongs to whereas the second belongs to which forces their intersection to lie on , the common line of the distinct planes , . Since this happens for all indices ̸ = we are finished with part (A).
The proof of part (B) is a bit longer. For = 1 the proposition holds vacuously. For = 2 its truth is immediate whenever at least one of the generalized lines (1)(1 ), (2)(2 ) is not a line, whereas if both of them are lines, they either coincide (both with the common line of and ) and the results hold trivially, or else they are distinct, in which case the four points (1), (1 ), (2), (2 ) do not lie on the same line. But then the lines (1)(2), (1 )(2 ) are also distinct, and as their intersection is nonempty by assumption, they define a plane on which lie the points (1), (1 ), (2), (2 ) forcing the lines (1)(1 ), (2)(2 ) to lie on this plane too and thus to have nonempty intersection (a single point, since these lines do not coincide).
For = 3 and assuming that at least one of the generalized lines (1)(1 ), (2)(2 ), (3)(3 ) is not a line, say (3)(3 ), the result comes from the truth of case = 2, since (1)(1 ), (2)(2 ) have to share some common point which of course lies on the generalized line (3)(3 ) as well. So assume that all three of these generalized lines are lines (Figure 1), which is the classical assumption for the converse of Desargues' Theorem.
Observe that at least one of the points (1 ), (2 ), (3 ), say (1 ), does not lie on ; otherwise the plane defined by these three noncollinear points would coincide with , which cannot happen. Then the line (1 )(2 ) does not lie on either, and so it is distinct from the line (1)(2) which lies on , and since (1 )(2 ), (1)(2 ) have a nonempty intersection, they define a plane, say 12

A Generalization for the Converse of Desargues' Theorem for Triangle-Connected Given Points
The assumptions postulated in part (B) of Proposition 1 can be considerably relaxed. For example, there is no reason for assuming that all lines ( )( ) intersect with their corresponding lines ( )( ), since a much smaller number of lines among the ( )( )'s with the same property force all the rest to behave similarly. This is made clear in Figure 5 where the lines of the sides of triangle (1)(2) This argument extends similarly for any ≥ 3 given points ( ) and ( ) on the two planes. All we need is to assume that all points ( ) that do not coincide with their corresponding points ( ) are similarly triangle-connected on the two planes. The meaning of this should be rather clear.
A triangle-connection sequence in or for simplicity just triangle-connection sequence for two points , of a given set will be any finite sequence of triangles formed by lines through the points of so that any two consecutive triangles share a common side and belongs to the first triangle of this sequence whereas belongs to the last one; recall from Section 1 that a triangle is a set of three lines. A triangleconnection set for will be a set of triangle-connection sequences of points of , that contains exactly one such sequence for each pair of points of . Of course whenever such a set exists for , it is not necessarily unique. The triangles in the sequences of such a set will be called connection triangles for and the lines of the sides connection lines. Two given sets of points (1), . . . , ( ) and (1 ), . . . , ( ) on two planes , where ( ) corresponds to ( ) will be called similarly triangleconnected whenever there exists a triangle-connection set for the ( )'s which becomes a triangle-connection set for the ( )'s by replacing all vertices of all triangles in the sequences of this set by their corresponding points on the other plane.
For example, a triangle-connection set for the set = {(1), (2), . . . , ( )} in Figure 6  The existence of the triangle-connection sets guarantees that ≥ 3 and that at least three points on each of the planes do not coincide with their corresponding points on the other. If exactly two points on each plane do not coincide with their corresponding points, then all but two lines ( )( ) are generalized but the remaining two are not guaranteed to have a nonempty intersection and thus this case cannot be included to the proposition.

Whenever Three or More Collinear Points Are Allowed but Corresponding Points Are Assumed Distinct
We close with a third generalization of the converse of Desargues' Theorem. In Proposition 1 the given points on the two planes were allowed to coincide with their corresponding points, but no three given points on any of the planes were allowed to be collinear. Flipping over these assumptions and restrictions, we now forbid any two corresponding points to coincide but allow three or more of them on any of the planes to be collinear. Figure 7 reveals that the converse of Desargues' Theorem does not always hold then; at least it does not hold whenever the given points on the planes are all collinear. Interestingly enough, it does hold whenever the given points on one of the planes do not all lie in a single line.