Classical Geometry: Euclidean, Transformational, Inverse and Projective

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CLASSICAL GEOMETRY: Euclidean, Transformational, Inversive, and Projective Copyright 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