1st Edition

College Geometry A Unified Development

By David C. Kay Copyright 2011
    652 Pages 657 B/W Illustrations
    by CRC Press

    Designed for mathematics majors and other students who intend to teach mathematics at the secondary school level, College Geometry: A Unified Development unifies the three classical geometries within an axiomatic framework. The author develops the axioms to include Euclidean, elliptic, and hyperbolic geometry, showing how geometry has real and far-reaching implications. He approaches every topic as a fresh, new concept and carefully defines and explains geometric principles.

    The book begins with elementary ideas about points, lines, and distance, gradually introducing more advanced concepts such as congruent triangles and geometric inequalities. At the core of the text, the author simultaneously develops the classical formulas for spherical and hyperbolic geometry within the axiomatic framework. He explains how the trigonometry of the right triangle, including the Pythagorean theorem, is developed for classical non-Euclidean geometries. Previously accessible only to advanced or graduate students, this material is presented at an elementary level. The book also explores other important concepts of modern geometry, including affine transformations and circular inversion.

    Through clear explanations and numerous examples and problems, this text shows step-by-step how fundamental geometric ideas are connected to advanced geometry. It represents the first step toward future study of Riemannian geometry, Einstein’s relativity, and theories of cosmology.

    Lines, Distance, Segments, and Rays
    Intended Goals
    Axioms of Alignment
    A Glimpse at Finite Geometry
    Metric Geometry
    Eves’ 25-Point Affine Geometry: A Model for Axioms 0–4
    Distance and Alignment
    Properties of Betweenness: Segments and Rays
    Coordinates for Rays
    Geometry and the Continuum
    Segment Construction Theorems

    Angles, Angle Measure, and Plane Separation
    Angles and Angle Measure
    Plane Separation
    Consequences of Plane Separation: The Postulate of Pasch
    The Interior of an Angle: The Angle Addition Postulate
    Angle Construction Theorems
    Consequences of a Finite Metric

    Unified Geometry: Triangles and Congruence
    Congruent Triangles: SAS Hypothesis
    A Metric for City Centers
    The SAS Postulate and the ASA and SSS Theorems
    Euclid’s Superposition Proof: An Alternative to Axiom 12
    Locus, Perpendicular Bisectors, and Symmetry
    The Exterior Angle Inequality
    Inequalities for Triangles
    Further Congruence Criteria
    Special Segments Associated with Triangles

    Quadrilaterals, Polygons, and Circles
    Quadrilaterals
    Congruence Theorems for Convex Quadrilaterals
    The Quadrilaterals of Saccheri and Lambert
    Polygons
    Circles in Unified Geometry

    Three Geometries
    Parallelism in Unified Geometry and the Influence of α
    Elliptic Geometry: Angle-Sum Theorem
    Pole-Polar Theory for Elliptic Geometry
    Angle Measure and Distance Related: Archimedes’ Method
    Hyperbolic Geometry: Angle-Sum Theorem
    A Concept for Area: AAA Congruence
    Parallelism in Hyperbolic Geometry
    Asymptotic Triangles in Hyperbolic Geometry
    Euclidean Geometry: Angle-Sum Theorem
    Median of a Trapezoid in Euclidean Geometry
    Similar Triangles in Euclidean Geometry
    Pythagorean Theorem

    Inequalities for Quadrilaterals: Unified Trigonometry
    An Inequality Concept for Unified Geometry
    Ratio Inequalities for Trapezoids
    Ratio Inequalities for Right Triangles
    Orthogonal Projection and "Similar" Triangles in Unified Geometry
    Unified Trigonometry: The Functions c(θ) and s(θ)
    Trigonometric Identities
    Classical Forms for c(θ) and s(θ)
    Lambert Quadrilaterals and the Function C(u)
    Identities for C(u)
    Classical Forms for C(u)
    The Pythagorean Relation for Unified Geometry
    Classical Unified Trigonometry

    Beyond Euclid: Modern Geometry
    Directed Distance: Stewart’s Theorem and the Cevian Formula
    Formulas for Special Cevians
    Circles: Power Theorems and Inscribed Angles
    Using Circles in Geometry
    Cross Ratio and Harmonic Conjugates
    The Theorems of Ceva and Menelaus
    Families of Mutually Orthogonal Circles

    Transformations in Modern Geometry
    Projective Transformations
    Affine Transformations
    Similitudes and Isometries
    Line Reflections: Building Blocks for Isometries and Similitudes
    Translations and Rotations
    Circular Inversion

    Non-Euclidean Geometry: Analytical Approach
    Law of Sines and Cosines for Unified Geometry
    Unifying Identities for Unified Trigonometry
    Half-Angle Identities for Unified Geometry
    The Shape of a Triangle in Unified Geometry: Cosine Inequality
    The Formulas of Gauss: Area of a Triangle
    Directed Distance: Theorems of Menelaus and Ceva
    Poincarè’s Model for Hyperbolic Geometry
    Other Models: Surface Theory
    Hyperbolic Parallelism and Bolyai’s Ideal Points

    Appendix A: Sketchpad Experiments
    Appendix B: Intuitive Spherical Geometry
    Appendix C: Proof in Geometry
    Appendix D: The Real Numbers and Least Upper Bound
    Appendix E: Floating Triangles/Quadrilaterals
    Appendix F: Axiom Systems for Geometry

    Solutions to Selected Problems

    Bibliography

    Index

    Biography

    Now retired, David C. Kay was a professor and chairman of the Department of Mathematics at the University of North Carolina–Asheville for 14 years. He previously taught at the University of Oklahoma for 17 years. His research interests include distance geometry, convexity theory, and related functional analysis.

    "The book is a comprehensive textbook on basic geometry. … Key features of the book include numerous figures and many problems, more than half of which come with hints or even complete solutions. Frequent historical comments add to making the reading a pleasant one."
    —Michael Joswig, Zentralblatt MATH 1273