Elementary Euclidean Geometry Research Articles

On Using GeoGebra and ChatGPT for Geometric Discovery

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results.

ПРЕПОДАВАНИЕ ГЕОМЕТРИИ В ПЕДАГОГИЧЕСКОМ ВУЗЕ НА ОСНОВАНИИ ГЕНЕТИЧЕСКОГО ПОДХОДА

The article considers the genetic approach to teaching mathematics and its application to teaching geometry. It involves the genetic development of important mathematical concepts, including the analysis of the subject from the historical, logical and epistemological, psychological and socio-cultural points of view, and the identification of logical genealogies of concepts and theorems. The importance of this type of approach to the method of teaching a basic course of geometry in the mathematical departments of pedagogical universities is substantiated within the framework of the Concept of Mathematical Education in the Russian Federation. The article deals with the applications of the genetic approach to teaching one of the sections of the geometry course, namely „Elements of Lobachevsky’s Geometry”, more specifically to its part, which deals with the statements equivalent to the axiom of parallelism of elementary Euclidean geometry.

The Many Sides of the Pythagorean Theorem

SummaryThe Pythagorean theorem gives a fundamental relationship between the lengths of the three sides of a right-angled triangle. We make the case for which theorems of elementary Euclidean geometry should be called the “Pythagorean theorem for an arbitrary triangle” and the “Pythagorean theorem for a quadrilateral.” In making this presentation, we stress the use of the basic properties of triangles over use of the cosine formula.

"Una mancha en el sol de Euclides": Hilbert y la teoría euclídea de las proporciones

El artículo examina una de las contribuciones más importantes a los fundamentos de la geometría euclídea elemental lograda por David Hilbert en su obra Fundamentos de la geometría (1899), a saber: la reconstrucción de la teoría euclídea de las proporciones y de los triángulos semejantes. Se argumenta que dicha reconstrucción no sólo estuvo motivada por la identificación de Hilbert de suposiciones implícitas en la teoría de Euclides, sino que además estuvo esencialmente ligada a la preocupación por la 'pureza del método'. Más aún, se afirma que, en este caso específico, el requerimiento de Hilbert por la pureza del método posee un carácter general o fundacional, esto es, no se refiere a la demostración de un teorema en particular, sino más bien es planteado respecto de la construcción axiomática de la teoría misma.

The statistical shape of geometric reasoning

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task – planar triangle completion. An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants’ estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our findings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

Classifying Voronoi graphs of hex spheres

A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2 π 3 but less than 2 π. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe the Voronoi regions of hex spheres and classify the Voronoi graphs of hex spheres (up to graph isomorphism).

Elementary Euclidean geometry: an introduction

This book, first published in 2004, is a genuine introduction to the geometry of lines and conics in the Euclidean plane. Lines and circles provide the starting point, with the classical invariants of general conics introduced at an early stage, yielding a broad subdivision into types, a prelude to the congruence classification. A recurring theme is the way in which lines intersect conics. From single lines one proceeds to parallel pencils, leading to midpoint loci, axes and asymptotic directions. Likewise, intersections with general pencils of lines lead to the central concepts of tangent, normal, pole and polar. The treatment is example based and self contained, assuming only a basic grounding in linear algebra. With numerous illustrations and several hundred worked examples and exercises, this book is ideal for use with undergraduate courses in mathematics, or for postgraduates in the engineering and physical sciences.

Ceva, Menelaus, and Selftransversality

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.

On the circles of Apollonius associated with a triangle

Given a ratio α ∶ β ∶ γ, α>γ>0, and a triangle ΔABC, on the sides\(\overline {BC} ,\overline {CA} \) and\(\overline {AB} \), using ratiosβ ∶ γ, α ∶ γ and α ∶ β, three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.

On certain linear mappings between inner-product and squared-distance matrices

We obtain the spectral decomposition of four linear mappings. The first, κ, is a mapping of the linear hull of all centered inner-product matrices onto the linear hull of all the induced squared-distance matrices. It is based on the natural generalization of the cosine law of elementary Euclidean geometry. The other three mappings studied are κ −1, the adjoint κ ∗, and (κ ∗) −1. Extensions and applications, particularly to multidimensional scaling, are discussed in some detail.

H. L. Royden. Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957-January 4, 1958, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 86–96.

H. L. Royden. Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957-January 4, 1958, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 86–96.

Dana Scott. Dimension in elementary Euclidean geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957–January 4,1958. Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 53–67.

Dana Scott. Dimension in elementary Euclidean geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957–January 4,1958. Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 53–67.

Some Concepts of Conformal Geometry

There are several introductions to conformal geometry in the French and German literatures. However, there does not seem to exist a recent one in English. The present article cannot claim to be one. We only wish to develop a few basic notions of n-dimensional conformal geometry and to elaborate its connection with the projective geometry on the n-sphere and with (n+2)dimensional pseudo-euclidean geometry. As far as the author knows, general conformal geometry over an arbitrary field F has not yet been discussed. By means of a (generalized) stereographic projection it can be related to the geometry on an arbitrary regular n-dimensional quadric in a projective (n+1)-space over F. In its turn this geometry is related to a (generalized) euclidean geometry in a centro-affine (n+2)-space over F. The bilinear form defining this geometry will be regular but not necessarily definite. The first two geometries possess homogeneous orthogonal groups while the group of the last one is inhomogeneous. The transition to this last geometry is based on the well-known idea of providing the intersections of the quadric with the linear n-spaces with norms. The new space elements [(n 1)quadrics plus norms] can be mapped one-one onto the vectors $0 of affine (n +2)-space. In the present note, only classical conformal geometry over the complex field is discussed. But the author has tried to use tools suitable for the more general situation indicated above. On the other hand he has also tried-in particular in the first sections-to motivate the introduction of conformal concepts from the point of view of elementary euclidean geometry. The approach chosen in this paper is based on the stereographic projection of (most of) euclidean n-space E. into the n-sphere in E,+ Insisting that this fundamental mapping be one-one, we are led in a natural way to imbed En into its conformal closure and E,+, into projective (n+1)-space. The algebraic expression for this projection is obtained in several leisurely steps. An impatient reader is referred to formulas (3.21), (3.31), (3.32). In the last sections inversions are discussed. They are shown to have most of the properties familiar from real 3-space. In particular the general conformal transformation can be decomposed into inversions. However, there are exceptions to Liouville's theorem in complex conformal spaces and not every conformal transformation will be the product of an inversion and a similarity.

Intelligent Behavior in Problem-Solving Machines

As one step in the study of intelligent behavior in machines, the authors consider the particular case of a machine that can prove theorems in elementary Euclidean plane geometry. The device uses no advanced decision algorithm, but relies rather on rudimentary mathematics and "ingenuity" in the manner, for example, of a clever high-school student. This paper discusses heuristic methods and learning machines and introduces the concept of a theory machine as an extension of a theorem-proving machine.

Intelligent Behavior in Problem-Solving Machines

As one step in the study of intelligent behavior in machines, the authors consider the particular case of a machine that can prove theorems in elementary Euclidean plane geometry. The device uses no advanced decision algorithm, but relies rather on rudimentary mathematics and “ingenuity” in the manner, for example, of a clever high-school student. This paper discusses heuristic methods and learning machines and introduces the concept of a theory machine as an extension of a theorem-proving machine.

The Complete Angle and Geometrical Generality

The very great importance to Elementary Euclidean Geometry of the figure intermediate between the Straight Line and the Triangle appears to have been missed: viz. of the plane figure formed by two intersecting straight lines, each unbounded both ways. For this type of figure we use the name “Complete Angle.”