Theorems of Euclidean Geometry Through Calculus

New Elements of Euclidean Geometry, 2021

This book introduces a new basis for Euclidean geometry consisting of 29 definitions, 10 axioms and 45 corollaries with which it is possible to prove the strong form of Euclid's First Postulate, Euclid's Second Postulate, Hilbert's axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, the axioms of Posidonius-Geminus, of Proclus, of Cataldi, of Tacquet 11, of Khayyam, of Playfair, Euclid's Postulate 5, and the historical statements proposed in place of Euclid's Postulate 5. The book ends with a chapter that culminates in the proof of the Pythagorean theorem and its converse.

Ezequias Cassela , 2021

This article aims to study the ellipse from the perspective of pure or synthetic geometry to the representation of points on a plane through the use of real numbers, as well as the representation and classification of this conic curve through the use of equations. The perspective developed in this article is based on the view of René Descartes, in considering that "the algebraic steps in a demonstration should really correspond to a geometric representation." The relevance of this article is to bring a reflection that eliminates the study of Analytical Geometry through ready-made and finished formulas, without satisfactory justification and without a logical chain that gives a greater meaning to the studied concepts. In general, the study developed in this article emphasizes the demonstration of results based on propositions adapted a priori, whose ability to be developed is aimed at establishing an "if...then" type of reasoning, making conjectures involving various knowledge already acquired and confirming such truths from a logical system, using definitions and propositions. Therefore, the demonstrations made in the scope of Synthetic Geometry will help to establish a connection with the equations obtained from the perspective of Analytical Geometry, serving as a consultation for students and professors of Analytical Geometry, thus avoiding sudden transitions between contents of degrees of distinct difficulties.

Mathematical Intelligencer, 2006

SYDNEY, WILD EGG, 2005 (http://wildegg.com) 300pp. AUD79.95 (about $60) HARDBACK ISBN 0-9757492-0-X

arXiv (Cornell University), 2021

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl's system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (i) it supports the thesis that Euclidean geometry is a priori, (ii) it supports the thesis that in modern mathematics the Weyl's system of axioms is dominant to the Euclid's system because it reflects the a priori underlying symmetries, (iii) it gives a new and promising approach to learn geometry which, through the Weyl's system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.

2020

An argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world. keywords: symmetry, Euclidean geometry, axioms, Weyl's axioms, philosophy of geometry Until the appearance of non-Euclidean geometries, Euclidean geometry and numbers had an equal status in mathematics. Indeed, until then, mathematics was described as the science of numbers and space. Whether it was thought that mathematical objects belong to a special world of ideas (Plato), or that they are ultimate abstractions drawn from the real world (Aristotle), or that they are a priori forms of our rational cognition (Kant), mathematical truths were considered, because of the clarity of their subject matter, a priori objective truths that are not subject to experimental veri cation. Descartes in Meditations (1641) writes: I counted as the most certain the truths which I concieved clearly as regards gures, numbers, an...

2004

Inverse and quotient of two vectors, 7.-Priority of algebraic operations, 8.-Geometric algebra of the vector plane, 9.-Exercises, 9. 2. A vector basis for the Euclidean plane Linear combination of two vectors, 10.-Basis and components, 10.-Orthonormal bases, 11.-Applications of formulae for products, 11.-Exercises, 12. 3. Complex numbers The subalgebra of complex numbers, 13.-Binomial, polar and trigonometric form of a complex number, 13.-Algebraic operations with complex numbers, 14.-Permutation of complex numbers and vectors, 17.-The complex plane, 18.-Complex analytic functions, 19.-Fundamental theorem of algebra, 24.-Exercises, 26. 4. Transformations of vectors Rotations, 27.-Axial symmetries, 28.-Inversions, 29.-Dilations, 30.-Exercises, 30 Second Part: Geometry of the Euclidean plane 5. Points and straight lines Translations, 31.-Coordinate systems, 31.-Barycentric coordinates, 33.-Distance between two points and area, 33.-Condition of collinearity of three points, 35.-Cartesian coordinates, 36.-Vectorial and parametric equations of a line, 36.-Algebraic equation and distance from a point to a line, 37.-Slope and intercept equations of a line, 40.-Polar equation of a line, 41.-Intersection of two lines and pencil of lines, 41.-Dual coordinates, 43.-Desargues's theorem, 48.-Exercises, 50. 6. Angles and elemental trigonometry Sum of the angles of a polygon, 53.-Definition of trigonometric functions and fundamental identities, 54.-Angle inscribed in a circle and double-angle identities, 55.-Addition of vectors and sum of trigonometric functions, 56.-Product of vectors and addition identities, 57.-Rotations and de Moivre's identity, 58.-Inverse trigonometric functions, 59.-Exercises, 60. 7. Similarities and simple ratio Direct similarity (similitude), 61.-Opposite similarity, 62.-Menelaus' theorem, 63.-Ceva's theorem, 64.-Homothety and simple ratio, 65.-Exercises, 67. 8. Properties of triangles Area of a triangle, 68.-Medians and centroid, 69.-Perpendicular bisectors and circumcentre, 70.-Angle bisectors and incentre, 72.-Altitudes and orthocentre, 73.-Euler's line, 76.-Fermat's theorem, 77.-Exercises, 78. XIII 9. Circles Algebraic and Cartesian equations, 80.-Intersections of a line with a circle, 80.-Power of a point with respect to a circle, 82.-Polar equation, 82.-Inversion with respect to a circle, 83.-The nine-point circle, 85.-Cyclic and circumscribed quadrilaterals, 87.-Angle between circles, 89.-Radical axis of two circles, 89.-Exercises, 91. 10. Cross ratios and related transformations Complex cross ratio, 92.-Harmonic characteristic and ranges, 94.-Homography (Möbius transformation), 96.-Projective cross ratio, 99.-Points at infinity and homogeneous coordinates, 102.-Perspectivity and projectivity, 103.-Projectivity as a tool for theorem demonstrations, 108.-Homology, 110.-Exercises, 115. 11. Conics Conic sections, 117.-Two foci and two directrices, 120.-Vectorial equation, 121.-Chasles' theorem, 122.-Tangent and perpendicular to a conic, 124.-Central equations for ellipse and hyperbola, 126.-Diameters and Apollonius' theorem, 128.-Conic passing through five points, 131.-Pencil of conics passing through four points, 133.-Conic equation in barycentric coordinates and dual conic, 133.-Polarities, 135.-Reduction of the conic matrix to diagonal form, 136.-Exercises, 137. Third part: Pseudo-Euclidean geometry 12. Matrix representation and hyperbolic numbers Rotations and the representation of complex numbers, 139.-The subalgebra of hyperbolic numbers, 140.-Hyperbolic trigonometry, 141.-Hyperbolic exponential and logarithm, 143.-Polar form, powers and roots of hyperbolic numbers, 144.-Hyperbolic analytic functions, 147.-Analyticity and square of convergence of power series, 150.-About the isomorphism of Clifford algebras, 152.-Exercises, 153. 13. The hyperbolic or pseudo-Euclidean plane Hyperbolic vectors, 154.-Inner and outer products of hyperbolic vectors, 155.-Angles between hyperbolic vectors, 156.-Congruence of segments and angles, 158.-Isometries, 158.-Theorems about angles, 160.-Distance between points, 160.-Area in the hyperbolic plane, 161.-Diameters of the hyperbola and Apollonius' theorem, 163.-The law of sines and cosines, 164.-Hyperbolic similarity, 167.-Power of a point with respect to a hyperbola with constant radius, 168.-Exercises, 169. Fourth part: Plane projections of three-dimensional spaces 14. Spherical geometry in the Euclidean space The geometric algebra of the Euclidean space, 170.-Spherical trigonometry, 172.-The dual spherical triangle of a given triangle, 175.-Right spherical triangles and Napier's rule, 176.-Area of a spherical triangle, 176.-Properties of the projections of the spherical surface, 177.-Central or gnomonic projection, 177.-Stereographic projection, 180.-Orthographic projection, 181.-Lambert's azimuthal equivalent projection, 182.-Spherical coordinates and cylindrical equidistant (plate carré) projection, 183.-Mercator XIV projection, 184.-Cylindrical equivalent projection, 184.-Conic projections, 185.-Exercises, 186. 15. Hyperboloidal geometry in the pseudo-Euclidean space The geometric algebra of the pseudo-Euclidean space, 189.-The hyperboloid of two sheets (Lobachevskian surface), 191.-Central projection (Beltrami disk), 192.-Lobachevskian trigonometry, 197.-Stereographic projection (Poincaré disk), 199.-Azimuthal equivalent projection, 201.-Weierstrass coordinates and cylindrical equidistant projection, 202.-Cylindrical conformal projection, 203.-Cylindrical equivalent projection, 204.-Conic projections, 204.-About the congruence of geodesic triangles, 206.-The hyperboloid of one sheet, 206.-Central projection and arc length on the one-sheeted hyperboloid, 207.-Cylindrical projections, 208.-Cylindrical central projection, 209.-Cylindrical equidistant projection, 210.-Cylindrical equivalent projection, 210.-Cylindrical conformal projection, 210.-Area of a triangle on the onesheeted hyperboloid, 211.-Trigonometry of right triangles, 214.-Hyperboloidal trigonometry, 215.-Dual triangles, 218.-Summary, 221.-Comment about the names of the non-Euclidean geometry, 222.-Exercises, 222. 16. Solutions to the proposed exercises 1. Euclidean vectors and their operations, 224.-2. A vector basis for the Euclidean plane, 225.-3. Complex numbers, 227.-4. Transformations of vectors, 230.-5. Points and straight lines, 231.-6. Angles and elemental trigonometry, 241.-7. Similarities and simple ratio, 244.-8. Properties of triangles, 246.-9. Circles, 257.-10. Cross ratios and related transformations, 262.-11. Conics, 266.-12. Matrix representation and hyperbolic numbers, 273.-13. The hyperbolic or pseudo-Euclidean plane, 275.-14. Spherical geometry in the Euclidean space, 278.-15. Hyperboloidal geometry in the pseudo-Euclidean space, 284. Bibliography, 293. Internet bibliography, 296. Index, 298. Chronology of the geometric algebra, 305.

Revue de Synthèse, 2003

Enriques, et autres-ont joué un rôle important dans la discussion sur les fondements des mathématiques. Mais, contrairement aux idées d'Euclide, ils n'ont pas identifié « l'espace physique » avec « l'espace de nos sens ». Partant de notre expérience dans l'espace, ils ont cherché à identifier les propriétés les plus importantes de l'espace et les ont posées à la base de la géométrie. C'est sur la connaissance active de l'espace que les axiomes de la géométrie ont été élaborés ; ils ne pouvaient donc pas être a priori comme ils le sont dans la philosophie kantienne. En outre, pendant la dernière décade du siècle, certains mathématiciens italiens-De Paolis, Gino Fano, Pieri, et autres-ont fondé le concept de nombre sur la géométrie, en employant des résultats de la géométrie projective. Ainsi, on fondait l'arithmétique sur la géométrie et non l'inverse, comme David Hilbert a cherché à faire quelques années après, sans succès.

1995

This review volume consists of articles by outstanding scientists who explore Archimedes' influence on the development of mathematics, particularly on Geometry, Analysis and Mechanics.

2024

This article presents an introduction to a new logical reasoning method for Euclidean geometry based on the invariance principle of information (data). Due to this principle, any geometrical configuration (shape or entity) in Euclidean geometry possesses a conserved content of geometrical data that could be expressed as information bits. On the basis of this principle, we explain what is the content of information bits and how is the analysis of this content in Euclidean geometry. Every geometric structure is actually a set of information bits that define that structure. We discuss the relationship between bits of information with the use of straight edge and compass in Euclidean geometry and show that theorems of Euclidean geometry can be expressed in the term of bits of information. In this regard, we show that the ability to draw a geometric structure, as well as theorems and propositions related to these structures, can be reduced to the rules of the invariance of information bits, and consequently leads to a new reasoning method in Euclidean geometry that can be applied for reasoning algorithms in AI. Based on this principle the converse of corresponding angles postulate and some of the theorems of constructible numbers and regular polygons are proved with concise proofs. In the field of number theory, by using prime number theorem, it is shown that measure of bit content of integer number n is equal to prime omega function Ω (n) and Shannon information (entropy) at the limit n → ∞.