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 a
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(¿) and s(¿)
Trigonometric Identities
Classical Forms for c(¿) and s(¿)
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
