On the Menelaus and Ceva 6-Figures in the Fibered Projective Planes

Abstract

The fibered versions of Menelaus and Ceva 6-figures in the fibered projective plane are given and the conditions to the fibered versions of Menelaus and Ceva 6-figures in the fibered projective plane with base plane, that is, projective plane, are determined.

1. Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy set. Since then, research on the theory and applications of fuzzy sets has been growing steadily. Several basic geometric concepts have been considered by various authors in a fuzzy context [2–4]. An attempt to develop a kind of fuzzy projective geometry deduced from fuzzy vector space was made by Kuijken et al. [5]. This provided a link between the fuzzy versions of classical theories that are very closely related. In these papers, for the most suitable definition of fuzzy geometry, they encountered the following problem. There are two possibilities to fuzzify the points of geometry. Either one assigns a unique degree of membership to each of them or one assigns several degrees of membership to all of them. They explored the second possibility, and it turns out that the geometric structures involved are much richer and defined fibered projective planes and looked at fibered versions of theorems of Desargues and Pappus [6]. The role of the triangular norm in the theory of fibered projective planes and fibered harmonic conjugates and a fibered version of Reidemeister’s condition were given in [7]. The authors considered different triangular norms in the theory of fibered projective planes. When the minimum operator is used, this implies that if the underlying projective plane is finite, then a finite number of -points generate a finite fibered projective plane. However, for other triangular norms, the fibered plane may be infinite. For instance, if we take , the ordinary product, then we see that, as soon as at least one membership degree is different from 1, we obtain infinitely many membership degrees and therefore infinitely many -points. Another observation to be made is that not all triangular norms are suitable here. For instance, if we take the Lukasiewicz norm, given by , then no point can have a membership degree different from 1. Hence a fibered projective plane with the Lukasiewicz triangular norm is equivalent to a crisp projective plane.

Kaya and Çiftçi introduced the 6-figures of Menelaus and Ceva in Moufang projective planes in [8]. In 1992, in order to state the theorems of Menelaus and Ceva, Klamkin and Liu needed the notion of signed distance in projective plane . In 2008, Kelly [9] generalized the Klamkin and Liu [10] result to projective planes , where is the field of characteristic not equal to two. But, in an arbitrary field, distance is not defined. Thus he used the scalars that are associated with the cross-ratio to prove these theorems in projective planes .

In this paper, since the definition of cross-ratio has not extended to fibered projective plane yet, we propose to contribute to fuzzy geometry by looking at the fibered version of Menelaus and Ceva 6-figures in the fibered projective plane which are given and the conditions to the fibered version of Menelaus and Ceva 6-figures in the fibered projective plane with base plane, that is, projective plane, are determined. We believe that such a concept will play an important role to establish a genuine fuzzy geometry.

2. Preliminaries

We first recall some basic notions from fuzzy set theory and fibered geometry. We denote by a triangular norm on the (real) unit interval , that is, a symmetric and associative binary operator satisfying whenever and , and , for all .

Definition 1 (see [6]). Let be any projective plane with point set and line set ; that is, and are two disjoint sets endowed with a symmetric relation (called the incidence relation) such that the graph is a bipartite graph with classes and and such that two distinct points , in are incident with exactly one line; every two distinct lines , are incident with exactly one point, and every line is incident with at least three points. A set of collinear points is a subset of , each member of which is incident with a common line . Dually, one defines a set of concurrent lines. We now define fibered points and fibered lines, which are briefly called -points and -lines.

Definition 2 (see [6]). Suppose that and . Then an -point is the following fuzzy set on the point set of : Dually, one defines in the same way the -line for and . The real number above is called the membership degree of the -point , while the point is called the base point of it, the same for -lines.

Definition 3 (see [6]). The two -lines and , with , intersect in the unique -point . Dually, the -points and , with , span the unique -line .

Definition 4 (see [6]). A (nontrivial) fibered projective plane consists of a set of -points of and a set of -lines of such that every point and every line of are the base point and base line of at least one -point and -line, respectively (with at least one membership degree different from ), and such that is closed undertaking intersections of -lines and spans of -points. Finally, a set of -points are called collinear if each pair of them span the same -line. Dually, a set of -lines are called concurrent if each pair of them intersects in the same -point.

Theorem 5 (see [6]). Suppose that one has a fibered projective plane with base plane , that is, Desarguesian. Choose the three -points , and in with noncollinear base points, and the three other -points , , and with noncollinear base points, such that the lines , for , are concurrent in a point of , with . Then the three -lines and (for and ) intersect in three collinear -points.

3. Fibered Version of Menelaus and Ceva 6-Figures

The Alexandrian Greek mathematician Menelaus and the seventeenth-century Italian mathematician Ceva are invariably mentioned together. Menelaus’ theorem, which involves a test for the collinearity of three points, and Ceva’s theorem, which involves a test for the concurrency of three lines, are frequently called the twin theorem. These theorems should have been discovered together, and it is not insignificant that such a long period separates Menelaus and Ceva. During the 1500 years that separate the two there was little development in mathematics. About the year A.D. 100, Menelaus of Alexandria extended a then well-known lemma to spherical triangles in his Sphaerica, the high point of Greek trigonometry. It is this lemma for the plane that today bears the name of Menelaus. Of the Ceva brothers, the lesser known Tommaso wrote on the cycloid while Giovanni resurrected the forgotten Menelaus’ theorem and published it in 1678 along with the twin theorem now known as Ceva’s theorem.

We now recall some observations regarding the triangular norm and the Menelaus and Ceva 6-figures.

In [7], the following theorems give an answer to the question, when does a finite fibered projective plane exist given a particular triangular norm?

Theorem 6 (see [7]). Let be a finite crisp projective plane and a triangular norm. Then there exists some finite nontrivial fibered projective plane with base plane if and only if there exists an idempotent element ; that is, .

Theorem 7 (see [7]). Let be an arbitrary crisp projective plane and a triangular norm. Then there exists some nontrivial fibered projective plane with base plane if and only if one of the following holds:(i)there exists an idempotent element ;(ii)there exists an element , with the property that , for all .

In [7], the definition and theorems related to -harmonic conjugate in a fibered projective plane whose base plane satisfies Fano’s axiom are given.

Definition 8 (see [7]). Suppose that one has a fibered projective plane with base projective plane . Choose the four -points , , , and in , none of the three base points of which are collinear. These -points are called -vertices. The configuration that consists of these four -points, the six -lines , for , (which we call -sides), and the three -points , with (the -diagonal points), is called an -complete quadrangle.

Theorem 9 (see [7]). Suppose that one has a fibered projective plane with base plane . Let , , and be three collinear points in and let , , be three -points of . Suppose that the point on obtained by intersecting with the join of the two diagonal points different from in any complete quadrangle, where are vertices and is a diagonal point, is independent of the chosen quadrangle. Then the -point obtained by intersecting with the -join of the two -diagonal points different from in any -complete quadrangle, where , are -vertices and is an -diagonal point, is independent of chosen -complete quadrangle.

The -point of previous theorem, if it exists, is called the fourth -conjugate to (, , ).

Theorem 10 (see [7]). Given a triangular norm , then is the minimum operator if and only if, in any fibered projective plane ,  for every quadruple (, , , ) of -points with collinear base points, the -point is the fourth -conjugate to (, , ) whenever is the fourth -conjugate to (, , ).

The original definitions of Menelaus and Ceva 6-figures are given in [11].

Definition 11 (see [8]). Let be a projective plane. A 6-figure in is a sequence of six distinct points such that constitutes a nondegenerate triangle with , , . The points , , , , , are called vertices of this 6-figure. Such a configuration is said to be a Menelaus 6-figure or a Ceva 6-figure if , , and are collinear or if , , are concurrent, respectively.

We now consider some classical configurations and theorems and extend them to fibered projective planes, for satisfying one of the conditions (i) and (ii) in Theorem 7, suitably.

Definition 12. Let be a fibered projective plane with base plane . Choose three -points , , in with noncollinear base points and the other three -points , , with for , . If the -points are -collinear, the configuration that consists of these six -points is called an -Menelaus 6-figure. It is called -Menelaus line spanned with -points for .

Theorem 13. Suppose that one has a fibered projective plane with base plane . Let , , be three noncollinear points in and let , , be three -points of . Suppose that the point on is obtained by intersecting with the join of two chosen points and where on and on . Then the -point obtained by intersecting with the -join of the two -points and , where , , and , , are -collinear, is independent of the chosen -points and .

Proof. Since the three -points , , and the three -points , , are -collinear, and . One calculates that , and hence the -point is independent of the choice of the -points and .

Theorem 14. Suppose that one has a fibered projective plane with base plane . Choose the three -points , , and in , neither of the three base points of which is collinear. Let the -point be obtained by intersecting with the -join of the two -points and , where , , and , , are -collinear. Then the configuration that consists of the six -points , , , is an -Menelaus 6-figure.

Proof. Since the three -points , , and are -collinear, from Definition 12, the configuration that consists of the six -points , , , is an -Menelaus 6-figure.

Theorem 15. Let be a fibered projective plane with base plane . Choose three -points , , in with noncollinear base points and with for , . If , , and in are collinear, then the three -points , , are -collinear if and only if .

Proof. A configuration is picked such that three -points , , and and for , . Suppose the three -points , , are -collinear. Since three -points are -collinear and the three -points , , and , for , , are -collinear and . Then clearly .
Conversely, if are satisfied, . Then three -points , , and are -collinear.

The Fibered Menelaus Condition (FMC). Let be a fibered projective plane with base plane . Choose three -points , , in with noncollinear base points and the other three -points , , with for, . The configuration that consists of these six -points is Menelaus 6-figure if and only if .

Theorem 16. Let be a triangular norm. Let be any nontrivial fibered projective plane with base plane and triangular norm . Then there is at least triangular norm such that satisfies FMC.

Proof. When the minimum operator for is used, it implies that , , and are idempotent elements in Theorem 15. So FMC is satisfied. Clearly, using the operator defined by for all , and , otherwise, is a triangular norm, then also FMC is satisfied.

Pappus’ theorem is an important theorem in geometry. Firstly, Kuijken and Van Maldeghem gave the fibered version of Pappus’ theorem using the minimum operator for , [6].

It is known that Pappus’ theorem can be proved by using the Menelaus theorem. Now, we give the fibered version of Pappus’ theorem by using the -Menelaus 6-figure.

Theorem 17. Let be a fibered projective plane with base plane . The fibered version of the Pappus configuration is obtained by using five -Menelaus configurations.

Proof. We choose two triples of -points , , and with collinear base points for , and such that neither of the three of the base points , , , is collinear. Then three intersection -points are , , and . Let , , and . If the triples of -collinear points , , , , and are -Menelaus 6-figure with the three -points , , , then , , , , and . It is easily seen that . The triple of -points is -Menelaus 6-figure with the three -points , , .

Definition 18. Let be a fibered projective plane with base plane . Choose three -points , , in with noncollinear base points and the other three -points , , with for , . If the -lines , , are -concurrent, the configuration that consists of these six -points is called an -Ceva 6-figure. The intersection point of the -lines , , is called -Ceva point.

Theorem 19. Suppose that one has a fibered projective plane with base plane . Let , , be three noncollinear points in and let , , be three -points of . Let points and be chosen such that on and on . Suppose that the point on is obtained by intersecting with the join and . Then the -point obtained by intersecting with the -join of the two -points and , where , , and , , are -collinear, is independent of the chosen -points and .

Proof. Since three -points , , and three -points , , are -collinear, and . One calculates that , and hence the -point is independent of the chosen -points and .

Theorem 20. Let be a fibered projective plane with base plane . Choose three -points , in with noncollinear base points and with for , . If the lines , and in are concurrent, then three -lines ,  for , are -concurrent if and only if .

Proof. A configuration is picked such that three -points , , and and for , . Suppose that three -lines , for , are -concurrent. Then three membership degrees in -concurrent point , , , are equal. Since three -points , for , , are valid, one can easily get .
Conversely, by using -points , , and that are collinear for , , in , it is shown that the three values , , , are equal.

Corollary 21 (the fibered Ceva condition (FCC)). Let be a fibered projective plane with base plane . Choose three -points , in with noncollinear base points and the other three -points , with for, . The configuration that consists of these six -points is Ceva 6-figure if and only if .

The following theorem shows that -Ceva 6-figures can be obtained as a corollary of -Menelaus 6-figures.

Theorem 22. Let be a fibered projective plane with base plane . Choose three -points , , in with noncollinear base points and the other three -points , , with for , ; the three lines are concurrent in . If the configuration that consists of these six -points is -Menelaus 6-figure, it is -Ceva 6-figure.

Proof. Let the configuration picked, such that the three -points , , and and for , , be Menelaus 6-figure. Three membership degrees in -intersection point of three -lines are , , . It is easily seen that these are equal. So three -lines for are -concurrent.

The reverse of this theorem is not true in .

Fano projective plane, denoted by , consists of seven points and seven lines. Fano projective plane is only example that is both Menelaus 6-figure and Ceva 6-figure. Even if the base plane of is Fano plane, the reverse of the process is not always valid in .

Theorem 23. Let be a triangular norm. Let be any nontrivial fibered projective plane with base plane , that is, Fano plane. Let three -points be , , in with noncollinear base points and the other three -points , , with for , ; the three lines are concurrent in . If the configuration that consists of these six -points is -Ceva 6-figure, it cannot be -Menelaus 6-figure.

Proof. The configuration is picked such that -points , , and , , is -Ceva 6-figure in . But, using the minimum operator for , it is easily seen that the -points , , and are not -collinear.

4. Conclusion

In this paper, the fibered versions of Menelaus and Ceva 6-figures in the fibered projective plane are defined. Although the conditions of the fibered versions of Menelaus and Ceva 6-figures are similar, the fibered versions of some of their properties in base projective plane cannot hold in fibered projective plane. -Ceva 6-figure is obtained from -Menelaus 6-figure automatically. Fano projective plane is both Menelaus 6-figure and Ceva 6-figure. But -Menelaus 6-figure cannot be obtained from -Ceva 6-figure in fibered projective plane with Fano plane. We have seen that the triangular norms have important role in the fiber versions of theorems related to theory.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.