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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.

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root., 28 pages, 10 figures

This paper provides a novel proof of a generalization of Pappus’ centroid theorem on [math] -dimensional tubes using Stokes’ theorem on manifolds.

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.

Pappusov teorem o šesterovrhu jedan je od najvažnijih teorema projektivne geometrije. On opisuje konfiguraciju od devet točaka i devet pravaca koja nastaje tako što se parovi nasuprotnih stranica šesterovrha, čiji vrhovi leže naizmjence na dva različita pravca, sijeku u trima kolinearnim točkama. U radu je izloženo osam različitih dokaza Papposovog teorema, s ciljem da se upoznaju i usporede različite metode pristupa. Posebno se razmatraju prednosti projektivno-geometrijskog pristupa u odnosu na osnovnu formulaciju teorema u okviru euklidske ravnine. Istaknuta je veza s Pascalovim teoremom i, preko Cayley-Bachrach-Chaslesovog teorema, dublja povezanost s algebarskom geometrijom. S druge strane, projektivnom interpretacijom euklidskog pojma kružnice pomoću para imaginarnih točaka na neizmjerno dalekom pravcu, dobiva se Miquelov teorem iz geometrije kružnica. Završno, naznačena je mogućnost formalnog algebarskog dokaza, preko homogenih koordinata, kojim se Pappusov teorem obuhvaća u svojoj punoj općenitosti. The hexagon theorem of Pappus is one of the most important theorems in projective geometry. It describes the configuration of nine lines and nine points which is generated by pairs of opposite sides of the hexagon, whose vertices lie alternatively on two different lines, intersect in three collinear points. In this paper we present eight proofs of Pappus’s theorem, with the aim of learning about different methods of approach to the problem and making a comparison between them. In particular, we expose the advantage of the projective geometry setting over the initial Euclidean formulation. We emphasize the connection to the Pascal’s theorem and, through the Cayley-Bachrach-Chasles theorem, the profound connection with Algebraic geometry. On the other hand, applying the projective interpretation of a circle, which is basically a Euclidean concept, by using a pair of imaginary points on the line at infinity, we derive Miquel’s theorem from the geometry of circles. Finally, we indicate the possibility of a formal algebraic proof using homogeneous coordinates, which encompasses Pappus’s theorem in its full generality.

This diploma thesis is focused on Selected problems in planimetry. The aim of this diploma thesis is description not only planimetric problems and their verification in a dynamic mathematical program GeoGebra, but also presentation of the author after whom it is called. The thesis is illustrated with pictures, which can help the reader to understand the problem and verification. This thesis can be used as a supplement the curriculum in secondary schools, where using dynamic program GeoGebra and subsequent verification may reach a better understanding of the topic.

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.

We prove a truncated second main theorem in the projective plane for entire curves which cluster on an algebraic curve.

We mainly give a brief survey on some methods and results related to Griffiths–Lang's conjecture on the second main theorem of holomorphic curves type into projective varieties. Also presented is the analogue of Cartan's theorem in number theory by Cartan's method, which is a refined form of Schmidt's subspace theorem.

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.

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. Then all the points in S are collinear. We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly's complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a 3-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem., Comment: 23 pages, 3 figures

We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:Let S be a finite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia) the colour different from k and is collinear with A and B. Then all the points in S are collinear.This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.

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.

We demonstrate that Pascal's classical theorem for hexagons inscribed in conics allows one to define in a compact manner maps which are governed algebraically by the integrable discrete CKP equation. A theorem for conics on oriented triangulated surfaces is used to construct a well-posed Cauchy problem for these dCKP maps. Moreover, the same theorem is exploited to construct in a purely geometric manner a Backlund transformation for dCKP maps. Thus, the integrability of dCKP maps and their underlying nonlinear soliton equation is shown to be encoded in an incidence theorem of projective geometry.

In Geometric, right-angled triangle is One Of the two—dimensional plane having three sides With one Of its angle is 9C and whk-h is important to solve problems related to Geometry and sometimes in Other subiect os well. Some fundamental concept 'theorems Of triangles are required to solve such problems and such theorems cre Pythagoras theorern' Pythagoras theorern (ii0 Appollonius theorem Euclids theorem' and (v) Eucli&s 20 theorem (Altitude theorem).3 I n addition to these, the author attempted to develop a new theorem related to right-angled triangle (Maran's theorem of right-angled triangle). The new theorem have been discussed and proved With relevant examples.

In this paper we study plane conic geometry, particularly different representations of geometric constructions and relations in plane conic geometry, with Cayley and bracket algebras. We propose three powerful simplification techniques for bracket computation involving conic points, and an algorithm for rational Cayley factorization in conic geometry. The factorization algorithm is not a general one, but works for all the examples tried so far. We establish a series of elimination rules for various geometric constructions based on the idea of bracket-oriented representation and elimination, and an algorithm for optimal representation of the conclusion in theorem proving. These techniques can be used in any applications involving brackets and conics. In theorem proving, our algorithm based on these techniques can produce extremely short proofs for difficult theorems in conic geometry.

(2002). The Theorem of Pappus: A Bridge Between Algebra and Geometry. The American Mathematical Monthly: Vol. 109, No. 6, pp. 497-516.

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 construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

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.

A simple geometric proof of Ebisui's theorem, if two triangles A1A2A3 and B1B2B3 are perspective and C3 = A1B2∩A2B1, C1=A2B3∩A3B2, C2=A3B1∩A1B3 then A1A2 A3 and C1C2C3 are also perspective, is given, which is using Desargues's theorem and its converse. With the theorem and an additional theorem, a (154, 203) configuration can be constructed, which is transitive both on points and lines.

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.

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.

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.

An approach to discovering new theorems in geometry using numerical examples is discussed. Four classes of theorems related to Pascal conics have been discovered by this approach. Two of them are new.

In this work is presented a generalization of Poncelet’s theorem to n dimensions which is refered to as the full Poncelet’s theorem. The theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon, then there exists an (n−1)‐parameter family of polygons having the same property. A physical realization and a projective geometrical proof of this theorem are given. If all the reflecting quadrics coincide, the above theorem reduces to the n‐dimensional Poncelet’s theorem presented by Chang and Friedberg. The geometrical proof is a finite construction based on a preliminary theorem which extends Hart’s lemma. The full Poncelet’s theorem may thus be extended to projective geometries over most fields, including discrete ones.

The paper contains a generalization to the

In the spirit of the Erdos-Szekeres theorem of 1935 we prove some canonical Ramsey Theorems in the plane. In Theorem 1 we are concerned with the number of points in the interior of a convex n -gon. In Theorem 2 we consider a set of points in general position along with a function from the set of points into the plane and prove the existence of a certain canonical configuration. A one-dimensional analog of Theorem 2 is a reformulation of a known theorem concerning intervals on the real line.

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.