Classical Geometry: Euclidean, Transformational, Inverse and Projective

CLASSICAL GEOMETRY: Euclidean, Transformational, Inversive, and Projective
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CONTENTS
Preface
PART I EUCLIDEAN GEOMETRY
1 PART I EUCLIDEAN GEOMETRY Congruency
1.1 Introduction
1.2 Congruent Figures
1.3 Parallel Lines
1.3.1 Angles in a Triangle
1.3.2 Thales' Theorem
1.3.3 Quadrilaterals
1.4 More About Congruency
1.5 Perpendiculars and Angle Bisectors
1.6 Construction Problems
1.6.1 The Method of Loci
1.7 Solutions to Selected Exercises
1.8 Problems
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
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
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
4.6.1 Ceva's Theorem in the Extended Plane
4.6.2 Menelaus' Theorem in the Extended Plane
4.7 Problems
5 Area
5.1 Basic Properties
5.1.1 Areas of Polygons
5.1.2 Finding the Area of Polygons
5.1.3 Areas of Other Shapes
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
6 Miscellaneous Topics
6.1 The Three Problems of Antiquity
6.2 Constructing Segments of Specific Lengths
6.3 Construction of Regular Polygons
6.3.1 Construction of the Regular Pentagon
6.3.2 Construction of Other Regular Polygons
6.4 Miquel's Theorem
6.5 Morley's Theorem
6.6 The Nine-Point Circle
6.6.1 Special Cases
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
7 The Euclidean Transformations or lsometries
7.1 Rotations, Reflections, and Translations
7.2 Mappings and Transformations
7.2.1 Isometries
7.3 Using Rotations, Reflections, and Translations
7.4 Problems
8 The Algebra of lsometries
8.1 Basic Algebraic Properties
8.2 Groups of Isometries
8.2.1 Direct and Opposite Isometries
8.3 The Product of Reflections
8.4 Problems
9 The Product of Direct lsometries
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
10 Symmetry and Groups
10.1 More About Groups
10.1.1 Cyclic and Dihedral Groups
10.2 Leonardo's Theorem
10.3 Problems
11 Homotheties
11.1 The Pantograph
11.2 Some Basic Properties
11.2.1 Circles
11.3 Construction Problems
11.4 Using Homotheties in Proofs
11.5 Dilatation
11.6 Problems
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 Ill INVERSIVE AND PROJECTIVE GEOMETRIES
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
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 Conics
14.5 Problems
15 Cross Ratios
15.1 Cross Ratios
15.2 Applications of Cross Ratios
15.3 Problems
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