Classical Geometry

I. E. Leonard is the author of Classical Geometry: Euclidean, Transformational, Inversive, and Projective Set, published by Wiley. J. E. Lewis is the author of Classical Geometry: Euclidean, Transformational, Inversive, and Projective Set, published by Wiley.

• Preface vPART I EUCLIDEAN GEOMETRY1 Congruency 31.1 Introduction 31.2 Congruent Figures 61.3 Parallel Lines 121.3.1 Angles in a Triangle 131.3.2 Thales' Theorem 141.3.3 Quadrilaterals 171.4 More About Congruency 211.5 Perpendiculars and Angle Bisectors 241.6 Construction Problems 281.6.1 The Method of Loci 311.7 Solutions to Selected Exercises 331.8 Problems 382 Concurrency 412.1 Perpendicular Bisectors 412.2 Angle Bisectors 432.3 Altitudes 462.4 Medians 482.5 Construction Problems 502.6 Solutions to the Exercises 542.7 Problems 563 Similarity 593.1 Similar Triangles 593.2 Parallel Lines and Similarity 603.3 Other Conditions Implying Similarity 643.4 Examples 673.5 Construction Problems 753.6 The Power of a Point 823.7 Solutions to the Exercises 873.8 Problems 904 Theorems of Ceva and Menelaus 954.1 Directed Distances, Directed Ratios 954.2 The Theorems 974.3 Applications of Ceva's Theorem 994.4 Applications of Menelaus' Theorem 1034.5 Proofs of the Theorems 1154.6 Extended Versions of the Theorems 1254.6.1 Ceva's Theorem in the Extended Plane 1274.6.2 Menelaus' Theorem in the Extended Plane 1294.7 Problems 1315 Area 1335.1 Basic Properties 1335.1.1 Areas of Polygons 1345.1.2 Finding the Area of Polygons 1385.1.3 Areas of Other Shapes 1395.2 Applications of the Basic Properties 1405.3 Other Formulae for the Area of a Triangle 1475.4 Solutions to the Exercises 1535.5 Problems 1536 Miscellaneous Topics 1596.1 The Three Problems of Antiquity 1596.2 Constructing Segments of Specific Lengths 1616.3 Construction of Regular Polygons 1666.3.1 Construction of the Regular Pentagon 1686.3.2 Construction of Other Regular Polygons 1696.4 Miquel's Theorem 1716.5 Morley's Theorem 1786.6 The Nine-Point Circle 1856.6.1 Special Cases 1886.7 The Steiner-Lehmus Theorem 1936.8 The Circle of Apollonius 1976.9 Solutions to the Exercises 2006.10 Problems 201PART II TRANSFORMATIONAL GEOMETRY7 The Euclidean Transformations or Isometries 2077.1 Rotations, Reflections, and Translations 2077.2 Mappings and Transformations 2117.2.1 Isometries 2137.3 Using Rotations, Reflections, and Translations 2177.4 Problems 2278 The Algebra of Isometries 2318.1 Basic Algebraic Properties 2318.2 Groups of Isometries 2368.2.1 Direct and Opposite Isometries 2378.3 The Product of Reflections 2418.4 Problems 2469 The Product of Direct Isometries 2539.1 Angles 2539.2 Fixed Points 2559.3 The Product of Two Translations 2569.4 The Product of a Translation and a Rotation 2579.5 The Product of Two Rotations 2609.6 Problems 26310 Symmetry and Groups 26910.1 More About Groups 26910.1.1 Cyclic and Dihedral Groups 27310.2 Leonardo's Theorem 27710.3 Problems 28111 Homotheties 28711.1 The Pantograph 28711.2 Some Basic Properties 28811.2.1 Circles 29111.3 Construction Problems 29311.4 Using Homotheties in Proofs 29811.5 Dilatation 30211.6 Problems 30412 Tessellations 31112.1 Tilings 31112.2 Monohedral Tilings 31212.3 Tiling with Regular Polygons 31712.4 Platonic and Archimedean Tilings 32312.5 Problems 330PART III INVERSIVE AND PROJECTIVE GEOMETRIES13 Introduction to Inversive Geometry 33713.1 Inversion in the Euclidean Plane 33713.2 The Effect of Inversion on Euclidean Properties 34313.3 Orthogonal Circles 35113.4 Compass-Only Constructions 36013.5 Problems 36914 Reciprocation and the Extended Plane 37314.1 Harmonic Conjugates 37314.2 The Projective Plane and Reciprocation 38314.3 Conjugate Points and Lines 39314.4 Conics 39914.5 Problems 40615 Cross Ratios 40915.1 Cross Ratios 40915.2 Applications of Cross Ratios 42015.3 Problems 42916 Introduction to Projective Geometry 43316.1 Straightedge Constructions 43316.2 Perspectivities and Projectivities 44316.3 Line Perspectivities and Line Projectivities 44816.4 Projective Geometry and Fixed Points 44816.5 Projecting a Line to Infinity 45116.6 The Apollonian Definition of a Conic 45516.7 Problems 461Bibliography 464Index 469