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We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Mobius, using hyperbolic geometry.

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.

We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a projective Butterfly theorem which covers all known affine cases.

Projective invariants are not only important objects in mathematics especially in geometry, but also widely used in many practical applications such as in computer vision and object recognition. In this work, we show a projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property, two high-dimensional generalizations of Pascal’s theorem are given, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex.

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.

This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras. The underlying framework for deriving the critical sets is projective geometry. Using a generalization of Pascal's Theorem, we prove that any number of cameras and scene points on an elliptic quartic form a critical set. Another important class of critical configurations consists of cameras and points on rational quartics. The theoretical results are accompanied by many examples and illustrations.

The Desargues theorem is a basic theorem in classical projective geometry. In this paper we generalize Desargues theorem in the direction of dynamical systems. Our result comprises an infinite family of configurations, having unbounded complexity. The proof of the result involves constructing special kinds of hyperplane arrangements and then projecting subsets of them into the plane.

By a transfer principle, Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed invariant theory of Leitenberger (J. Algebra222 (1999), 82), we construct corresponding quantum invariants by a computer calculation.

We discuss in the context of Euclidean and projective geometry a recently rediscovered theorem concerning the concurrency of the main diagonals of a hexagon, and provide a detailed proof of its hyperbolic version.

The purpose of this paper is to state and prove a theorem (the CMS Theorem) which generalizes the familiar Ceva's Theorem and Menelaus' Theorem of elementary Euclidean geometry. The theorem concernsn -acrons (generalizations of n-gons) in affine space of any number of dimensions and makes assertions about circular products of ratios of lengths, areas, volumes, etc. In particular it contains, as special cases, many results in this area proved by earlier authors.

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spacePn (n ≥2). Puttingn = 2, we showed in [3] that the theorem of Desargues inPn is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inPn. We only give a few examples and leave it as an open problem which other special cases could be found.

Thebault's problem was posed in 1938, but a solution did not appear until 1983. Since then other solutions of the original problem and of various generalisations have been given. In this paper we discuss the generalisations (including some new results) and various background theorems on tritangent centres, making extensive use of Pascal's theorem.

We present an algorithm that is able to confirm projective incidence statements by carrying out calculations in the ring of all formal determinants (brackets) of a configuration. We will describe an implementation of this prover and present a series of examples treated by the prover, includingPappus' andDesargues' theorems, thesixteen point theorem, Saam's theorem, thebundle condition, theuniqueness of a harmonic point andPascal's theorem.

We present a method which can produce traditional proofs for a class of constructive geometry statements in Euclidean geometry. The method is a mechanization of the traditional area method used by many geometers. The key idea of our method is to eliminate dependent (constructed) points in a geometry statement using a few basic geometry propositions about the area of triangles. The method has been implemented. Our program, calledEuclid, can produce traditional proofs of many hard geometry theorems such as Pappus' theorem, Pascal's theorem, Gauss point theorem, and the Pascal conic theorem. Currently, it has produced proofs of 110 nontrivial theoremsentirely automatically. The proofs produced byEuclid are elegant, short (often shorter than the proofs given by geometers) and understandable even to high school students. This method seems to be the first that can produce traditional proofs for hard geometry theorems automatically.

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration π, consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to π in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

H.S.M. Coxeter in his book “Introduction to Geometry” quotes a theorem of Mobius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.

A modified theorem of Mobius (see [1]) is considered. Its equivalence to the commutativity of the basic field of the projective space is proved.

In a projective plane the set of all points which are on homologous lines of two projective, non-perspective, different pencils with self-homologous common line is called a degenerated Steiner (d. S.) conic. An analytic description for d. S. conics is given. Amitsur's theorem about generalized polynomial identities in division rings finds two geometrical applications: (1) in the discussion of the normal form problem for d. S. conics and (2) in answering the question if a d. S. conic allows beside the defining generation further generations. We study the intersection of d. S. conics with lines and notice that a d. S. conic is never identical with a set of absolute points of a polarity.

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Beruhrsatz). The same special Pascal theorem now leads to an incidence theorem, π, for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of π contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.

Without using the representation theorem and a theorem of J.H.M. WEDDERBURN we show that every finite Desarguesian affine plane is Pappian.

Concerning Pappus lines and Leisenring lines there is a set of interesting theorems given in [1]. In this paper, we show that these theorems can be easily proved by our computer prover which is implemented on the basis of Wu's method for mechanical theorem proving in geometries. Using this prover we discovered, furthermore, a new theorem: The six intersection points of every Pappus line and its corresponding Leisenring line are collinear.

If S is a finite set of points in the plane and no conic contains all points of S, then S determines a conic which contains exactly five points ofS.

If on an oval in a projective plane a 4-point Pascal theorem, π, with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} ∪ (O) ∪ (∞), with c ≠ O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of π, then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. π may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of π characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.

An oval C In a projective plane is a set of triply noncollinear points such that each point of C is on exactly one line which contains no other point of C. An oval has the Pascalian property if each of its hexagons has collinear diagonal points. The author considers those hexagons in which at least two vertices coincide and shows that if every hexagon in this class of 5-point hexagons is Pascalian then the oval has the Pascalian property. Therefore, the 5-point Pascalian property is equivalent to the full Pascalian property. The proof makes use of the coordinatization found in Artzy [1, 2].

In [3] U. Ott introduced a Bruck-Ryser abstract theorem concerning ‘lattices of an R-module’. As one of the applications of this theorem, he obtained a new proof of the Bruck-Ryser theorem for finite projective planes and studied ‘p-curves’ in a projective plane. In this paper, we apply the Bruck-Ryser abstract theorem in order to give a new proof of the Bruck-Ryser-Chowla theorem for symmetric (v, k, λ)-design with (v, k, λ)=1, and to study ‘p-curves’ in arbitrary symmetric designs.