Classical Geometry – Euclidean, Transformational, Inversive, and Projective Set

<p>Preface v</p>
<p>PART I EUCLIDEAN GEOMETRY</p>
<p>1 Congruency 3</p>
<p>1.1 Introduction 3</p>
<p>1.2 Congruent Figures 6</p>
<p>1.3 Parallel Lines 12</p>
<p>1.3.1 Angles in a Triangle 13</p>
<p>1.3.2 Thales′ Theorem 14</p>
<p>1.3.3 Quadrilaterals 17</p>
<p>1.4 More About Congruency 21</p>
<p>1.5 Perpendiculars and Angle Bisectors 24</p>
<p>1.6 Construction Problems 28</p>
<p>1.6.1 The Method of Loci 31</p>
<p>1.7 Solutions to Selected Exercises 33</p>
<p>1.8 Problems 38</p>
<p>2 Concurrency 41</p>
<p>2.1 Perpendicular Bisectors 41</p>
<p>2.2 Angle Bisectors 43</p>
<p>2.3 Altitudes 46</p>
<p>2.4 Medians 48</p>
<p>2.5 Construction Problems 50</p>
<p>2.6 Solutions to the Exercises 54</p>
<p>2.7 Problems 56</p>
<p>3 Similarity 59</p>
<p>3.1 Similar Triangles 59</p>
<p>3.2 Parallel Lines and Similarity 60</p>
<p>3.3 Other Conditions Implying Similarity 64</p>
<p>3.4 Examples 67</p>
<p>3.5 Construction Problems 75</p>
<p>3.6 The Power of a Point 82</p>
<p>3.7 Solutions to the Exercises 87</p>
<p>3.8 Problems 90</p>
<p>4 Theorems of Ceva and Menelaus 95</p>
<p>4.1 Directed Distances, Directed Ratios 95</p>
<p>4.2 The Theorems 97</p>
<p>4.3 Applications of Ceva′s Theorem 99</p>
<p>4.4 Applications of Menelaus′ Theorem 103</p>
<p>4.5 Proofs of the Theorems 115</p>
<p>4.6 Extended Versions of the Theorems 125</p>
<p>4.6.1 Ceva′s Theorem in the Extended Plane 127</p>
<p>4.6.2 Menelaus′ Theorem in the Extended Plane 129</p>
<p>4.7 Problems 131</p>
<p>5 Area 133</p>
<p>5.1 Basic Properties 133</p>
<p>5.1.1 Areas of Polygons 134</p>
<p>5.1.2 Finding the Area of Polygons 138</p>
<p>5.1.3 Areas of Other Shapes 139</p>
<p>5.2 Applications of the Basic Properties 140</p>
<p>5.3 Other Formulae for the Area of a Triangle 147</p>
<p>5.4 Solutions to the Exercises 153</p>
<p>5.5 Problems 153</p>
<p>6 Miscellaneous Topics 159</p>
<p>6.1 The Three Problems of Antiquity 159</p>
<p>6.2 Constructing Segments of Specific Lengths 161</p>
<p>6.3 Construction of Regular Polygons 166</p>
<p>6.3.1 Construction of the Regular Pentagon 168</p>
<p>6.3.2 Construction of Other Regular Polygons 169</p>
<p>6.4 Miquel′s Theorem 171</p>
<p>6.5 Morley′s Theorem 178</p>
<p>6.6 The Nine–Point Circle 185</p>
<p>6.6.1 Special Cases 188</p>
<p>6.7 The Steiner–Lehmus Theorem 193</p>
<p>6.8 The Circle of Apollonius 197</p>
<p>6.9 Solutions to the Exercises 200</p>
<p>6.10 Problems 201</p>
<p>PART II TRANSFORMATIONAL GEOMETRY</p>
<p>7 The Euclidean Transformations or Isometries 207</p>
<p>7.1 Rotations, Reflections, and Translations 207</p>
<p>7.2 Mappings and Transformations 211</p>
<p>7.2.1 Isometries 213</p>
<p>7.3 Using Rotations, Reflections, and Translations 217</p>
<p>7.4 Problems 227</p>
<p>8 The Algebra of Isometries 231</p>
<p>8.1 Basic Algebraic Properties 231</p>
<p>8.2 Groups of Isometries 236</p>
<p>8.2.1 Direct and Opposite Isometries 237</p>
<p>8.3 The Product of Reflections 241</p>
<p>8.4 Problems 246</p>
<p>9 The Product of Direct Isometries 253</p>
<p>9.1 Angles 253</p>
<p>9.2 Fixed Points 255</p>
<p>9.3 The Product of Two Translations 256</p>
<p>9.4 The Product of a Translation and a Rotation 257</p>
<p>9.5 The Product of Two Rotations 260</p>
<p>9.6 Problems 263</p>
<p>10 Symmetry and Groups 269</p>
<p>10.1 More About Groups 269</p>
<p>10.1.1 Cyclic and Dihedral Groups 273</p>
<p>10.2 Leonardo′s Theorem 277</p>
<p>10.3 Problems 281</p>
<p>11 Homotheties 287</p>
<p>11.1 The Pantograph 287</p>
<p>11.2 Some Basic Properties 288</p>
<p>11.2.1 Circles 291</p>
<p>11.3 Construction Problems 293</p>
<p>11.4 Using Homotheties in Proofs 298</p>
<p>11.5 Dilatation 302</p>
<p>11.6 Problems 304</p>
<p>12 Tessellations 311</p>
<p>12.1 Tilings 311</p>
<p>12.2 Monohedral Tilings 312</p>
<p>12.3 Tiling with Regular Polygons 317</p>
<p>12.4 Platonic and Archimedean Tilings 323</p>
<p>12.5 Problems 330</p>
<p>PART III INVERSIVE AND PROJECTIVE GEOMETRIES</p>
<p>13 Introduction to Inversive Geometry 337</p>
<p>13.1 Inversion in the Euclidean Plane 337</p>
<p>13.2 The Effect of Inversion on Euclidean Properties 343</p>
<p>13.3 Orthogonal Circles 351</p>
<p>13.4 Compass–Only Constructions 360</p>
<p>13.5 Problems 369</p>
<p>14 Reciprocation and the Extended Plane 373</p>
<p>14.1 Harmonic Conjugates 373</p>
<p>14.2 The Projective Plane and Reciprocation 383</p>
<p>14.3 Conjugate Points and Lines 393</p>
<p>14.4 Conics 399</p>
<p>14.5 Problems 406</p>
<p>15 Cross Ratios 409</p>
<p>15.1 Cross Ratios 409</p>
<p>15.2 Applications of Cross Ratios 420</p>
<p>15.3 Problems 429</p>
<p>16 Introduction to Projective Geometry 433</p>
<p>16.1 Straightedge Constructions 433</p>
<p>16.2 Perspectivities and Projectivities 443</p>
<p>16.3 Line Perspectivities and Line Projectivities 448</p>
<p>16.4 Projective Geometry and Fixed Points 448</p>
<p>16.5 Projecting a Line to Infinity 451</p>
<p>16.6 The Apollonian Definition of a Conic 455</p>
<p>16.7 Problems 461</p>
<p>Bibliography 464</p>
<p>Index 469</p>