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