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.
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:
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
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 [
The general projective planes came to existence in an attempt to construct via incidence axioms planes in which 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: any two distinct points are incident with exactly one line; any two distinct lines are incident with exactly one point; 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 [
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 [
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 [ there exists exactly one line through any two distinct points; any two distinct lines on the same plane have a unique common point; there exist at least three points on each line; there exist three points not on the same line; there exist four points not on the same plane; there exists exactly one plane through any three distinct noncollinear points; there exists at least one point on each plane; if two distinct points are on a plane, then all points in the line containing the given points are on the same plane; if two planes have a common point, then they have at least one more common point. a plane and a line not on the plane always have a unique common point; two distinct planes intersect in a common line.
Some immediate corollaries are the following:
A number of given points will be called
Figure
No matter what the meaning of the line
So let
Desargues’ Theorem for
Let
(A) If all generalized lines
(B) If all the intersections of the pairs of lines
For
The proof of part (B) is a bit longer. For
For
Observe that at least one of the points
For
If there exist exactly two lines among the
Finally, if there exist at least three lines among the
Whenever the generalized lines
The above proof reveals that with the classical hypothesis for the converse of Desargues’ Theorem the three lines
The assumptions postulated in part (B) of Proposition
This is made clear in Figure
If
Also note that the points
This argument extends similarly for any
A
For example, a triangle-connection set for the set
(a) A triangle-connection set for the points
Now for two similarly triangle-connected sets of points
About the claim that for all
Let
The existence of the triangle-connection sets guarantees that
We close with a third generalization of the converse of Desargues’ Theorem. In Proposition
The converse of Desargues’ Theorem does not always hold in case all given points are collinear in both planes.
Let
For
Finally, for
If on the other hand no more than
The author declares that there is no conflict of interests regarding the publication of this paper.