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Classical Geometry: Euclidean, Transformational, Inversive, and Projective

Book Description

Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science

Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout.

The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes:

  • Multiple entertaining and elegant geometry problems at the end of each section for every level of study

  • Fully worked examples with exercises to facilitate comprehension and retention

  • Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications

  • An approach that prepares readers for the art of logical reasoning, modeling, and proofs

  • The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Preface
    6. Part I: Euclidean Geometry
      1. Chapter 1: Congruency
        1. 1.1 Introduction
        2. 1.2 Congruent Figures
        3. 1.3 Parallel Lines
        4. 1.4 More About Congruency
        5. 1.5 Perpendiculars and Angle Bisectors
        6. 1.6 Construction Problems
        7. 1.7 Solutions to Selected Exercises
        8. 1.8 Problems
      2. Chapter 2: Concurrency
        1. 2.1 Perpendicular Bisectors
        2. 2.2 Angle Bisectors
        3. 2.3 Altitudes
        4. 2.4 Medians
        5. 2.5 Construction Problems
        6. 2.6 Solutions to the Exercises
        7. 2.7 Problems
      3. Chapter 3: Similarity
        1. 3.1 Similar Triangles
        2. 3.2 Parallel Lines and Similarity
        3. 3.3 Other Conditions Implying Similarity
        4. 3.4 Examples
        5. 3.5 Construction Problems
        6. 3.6 The Power of a Point
        7. 3.7 Solutions to the Exercises
        8. 3.8 Problems
      4. Chapter 4: Theorems of Ceva and Menelaus
        1. 4.1 Directed Distances, Directed Ratios
        2. 4.2 The Theorems
        3. 4.3 Applications of Ceva’s Theorem
        4. 4.4 Applications of Menelaus’ Theorem
        5. 4.5 Proofs of the Theorems
        6. 4.6 Extended Versions of the Theorems
        7. 4.7 Problems
      5. Chapter 5: Area
        1. 5.1 Basic Properties
        2. 5.2 Applications of the Basic Properties
        3. 5.3 Other Formulae for the Area of a Triangle
        4. 5.4 Solutions to the Exercises
        5. 5.5 Problems
      6. Chapter 6: Miscellaneous Topics
        1. 6.1 The Three Problems of Antiquity
        2. 6.2 Constructing Segments of Specific Lengths
        3. 6.3 Construction of Regular Polygons
        4. 6.4 Miquel’s Theorem
        5. 6.5 Morley’s Theorem
        6. 6.6 The Nine-Point Circle
        7. 6.7 The Steiner-Lehmus Theorem
        8. 6.8 The Circle of Apollonius
        9. 6.9 Solutions to the Exercises
        10. 6.10 Problems
    7. Part II: Transformational Geometry
      1. Chapter 7: The Euclidean Transformations or Isometries
        1. 7.1 Rotations, Reflections, and Translations
        2. 7.2 Mappings and Transformations
        3. 7.3 Using Rotations, Reflections, and Translations
        4. 7.4 Problems
      2. Chapter 8: The Algebra of Isometries
        1. 8.1 Basic Algebraic Properties
        2. 8.2 Groups of Isometries
        3. 8.3 The Product of Reflections
        4. 8.4 Problems
      3. Chapter 9: The Product of Direct Isometries
        1. 9.1 Angles
        2. 9.2 Fixed Points
        3. 9.3 The Product of Two Translations
        4. 9.4 The Product of a Translation and a Rotation
        5. 9.5 The Product of Two Rotations
        6. 9.6 Problems
      4. Chapter 10: Symmetry and Groups
        1. 10.1 More About Groups
        2. 10.2 Leonardo’s Theorem
        3. 10.3 Problems
      5. Chapter 11: Homotheties
        1. 11.1 The Pantograph
        2. 11.2 Some Basic Properties
        3. 11.3 Construction Problems
        4. 11.4 Using Homotheties in Proofs
        5. 11.5 Dilatation
        6. 11.6 Problems
      6. Chapter 12: Tessellations
        1. 12.1 Tilings
        2. 12.2 Monohedral Tilings
        3. 12.3 Tiling with Regular Polygons
        4. 12.4 Platonic and Archimedean Tilings
        5. 12.5 Problems
    8. Part III: Inversive and Projective Geometries
      1. Chapter 13: Introduction to Inversive Geometry
        1. 13.1 Inversion in the Euclidean Plane
        2. 13.2 The Effect of Inversion on Euclidean Properties
        3. 13.3 Orthogonal Circles
        4. 13.4 Compass-Only Constructions
        5. 13.5 Problems
      2. Chapter 14: Reciprocation and the Extended Plane
        1. 14.1 Harmonic Conjugates
        2. 14.2 The Projective Plane and Reciprocation
        3. 14.3 Conjugate Points and Lines
        4. 14.4 Conies
        5. 14.5 Problems
      3. Chapter 15: Cross Ratios
        1. 15.1 Cross Ratios
        2. 15.2 Applications of Cross Ratios
        3. 15.3 Problems
      4. Chapter 16: Introduction to Projective Geometry
        1. 16.1 Straightedge Constructions
        2. 16.2 Perspectivities and Projectivities
        3. 16.3 Line Perspectivities and Line Projectivities
        4. 16.4 Projective Geometry and Fixed Points
        5. 16.5 Projecting a Line to Infinity
        6. 16.6 The Apollonian Definition of a Conic
        7. 16.7 Problems
    9. Bibliography
    10. Index