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Geometry Illuminated an Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry

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Geometry Illuminated an Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry AMS / MAA TEXTBOOKS VOL 30 Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry Matthew Harvey Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2015936098 Print ISBN: 978-1-93951-211-6 Electronic ISBN: 978-1-61444-618-7 Printed in the United States of America Current Printing (last digit): 10987654321 10.1090/text/030 Geometry Illuminated An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry Matthew Harvey The University of Virginia’s College at Wise Published and distributed by The Mathematical Association of America Council on Publications and Communications Jennifer J. Quinn, Chair Committee on Books Fernando Gouvea,ˆ Chair MAA Textbooks Editorial Board Stanley E. 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Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-240-396-5647 Contents Preface xv 0 Axioms and Models 1 0.1 Fano’s geometry .................................................................................. 2 0.2 Further reading ................................................................................... 4 0.3 Exercises........................................................................................... 5 I Neutral Geometry 7 1 The Axioms of Incidence and Order 9 1.1 Incidence........................................................................................... 11 1.2 Order................................................................................................ 12 1.3 Putting points in order .......................................................................... 14 1.4 Exercises........................................................................................... 16 1.5 Further reading ................................................................................... 17 2 Angles and Triangles 19 2.1 Exercises........................................................................................... 24 2.2 References......................................................................................... 25 3 Congruence Verse I: SAS and ASA 27 3.1 Triangle congruence............................................................................. 30 3.2 Exercises........................................................................................... 34 4 Congruence Verse II: AAS 37 4.1 Supplementary angles........................................................................... 37 4.2 The alternate interior angle theorem......................................................... 40 4.3 The exterior angle theorem .................................................................... 42 4.4 AAS................................................................................................. 43 4.5 Exercises........................................................................................... 44 vii viii Contents 5 Congruence Verse III: SSS 45 5.1 Exercises........................................................................................... 49 6 Distance, Length, and the Axioms of Continuity 51 6.1 Synthetic comparison ........................................................................... 51 6.2 Distance............................................................................................ 53 6.3 Exercises........................................................................................... 62 7 Angle Measure 63 7.1 Synthetic angle comparison.................................................................... 63 7.2 Right angles ....................................................................................... 67 7.3 Angle measure.................................................................................... 70 7.4 Exercises........................................................................................... 71 8 Triangles in Neutral Geometry 73 8.1 Exercises........................................................................................... 78 8.2 References......................................................................................... 79 9 Polygons 81 9.1 Definitions......................................................................................... 81 9.2 Counting polygons............................................................................... 83 9.3 Interiors and exteriors........................................................................... 84 9.4 Interior angles: two dilemmas................................................................. 88 9.5 Polygons of note.................................................................................. 92 9.6 Exercises........................................................................................... 93 10 Quadrilateral Congruence Theorems 95 10.1 Terminology....................................................................................... 95 10.2 Quadrilateral congruence....................................................................... 96 10.3 Exercises........................................................................................... 103 II Euclidean Geometry 105 11 The Axiom on Parallels 107 11.1 Exercises........................................................................................... 113 12 Parallel Projection 115 12.1 Parallel projection................................................................................ 116 12.2 Parallel projection, order, and congruence ................................................. 117 12.3 Parallel projection and distance............................................................... 120 12.4 Exercises........................................................................................... 124 13 Similarity 125 13.1 Triangle similarity theorems................................................................... 127 Contents ix 13.2 The Pythagorean theorem...................................................................... 130 13.3 Exercises........................................................................................... 133 14 Circles 135 14.1 Definitions......................................................................................... 135 14.2 Intersections....................................................................................... 137
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