College Geometry ─ A Unified Development

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.

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.

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