Geometry : A High School Course A geometry course based on this book was taught success­ fully by Gene Murrow for several years. We are much indebted to Springer Verlag for publishing Geometry so that others can try our approach. The publishers and we thought it would be appropriate to issue the book first in a preliml. nary edition on which we would welcome comments especially from students and teachers of the high school geometry course. Such comments can bear on any aspect of Geometry ranging from the choice of topics the ordering of the topics and other global considerations to possible computational errors and misprints. We shall welcome criticisms and suggestions. Serge Lang Gene Murrow Contents Theorems Proved in Geometry xi xvii Introduction CHAPTER 1 Distance and Angles 51. Lines 1 52. Distance 12 53. Angles 20 54. Proofs 43 55. Right Angles and Perpendicularity 52 86. The Angles of a Triangle 65 CHAPTER 2 Coordinates 51. Coordinate Systems 85 52. Distance between Points on a Line 94 53. Equation of a Line 96 CHAPTER 3 Area and the Pythagoras Theorem 51. The Area of a Triangle 107 S2. The Pythagoras Theorem 125 viii CONTENTS CHAPTER 4 The Distance Formula Sl. Distance between Arbitrary Points 142 S2. Higher Dimensional Space 148 S3. Equation of a Circle 155 CHAPTER 5 Some Applications of Right Triangles S1. Perpendicular Bisector 162 S2. Isosceles and Equilateral Triangles 175 S3. Theorems About Circles 190 CHAPTER 6 Polygons S1. 1 — Distance and Angles 2 — Coordinates 3 — Area and the Pythagoras Theorem 4 — The Distance Formula 5 — Some Applications of Right Triangles 6 — Polygons 7 — Congruent Triangles 8 — Dilations and Similarities 9 — Volumes 10 — Vectors and Dot Product 11 — Transformations 12 — Isometries.