EDITORIAL COMMITTEE Paul J. Sally, Jr. (Chair) Joseph Silverman Francis Su Susan Tolman For additional informationand updates on this book, visit www.ams.org/bookpages/amstext-21 Library of Congress Cataloging-in-Publication Data Lee,JohnM.,1950– Axiomaticgeometry/JohnM.Lee pagescm. —(Pureandappliedundergraduatetexts;volume21) Includesbibliographicalreferencesandindex. ISBN978-0-8218-8478-2(alk.paper) 1.Axioms. 2.Geometry. I.Title. Contents Preface....................................................................... xi Chapter1. Euclid............................................................ 1 ReadingEuclid............................................................. 2 AfterEuclid................................................................ 8 TheDiscoveryofNon-EuclideanGeometry................................... 10 GapsinEuclid’sArguments................................................. 13 ModernAxiomSystems..................................................... 20 Exercises.................................................................. 21 Chapter2. IncidenceGeometry ............................................... 23 AxiomaticSystems......................................................... 23 TheAxiomsofIncidenceGeometry.......................................... 24 InterpretationsandModelsofIncidenceGeometry............................. 26 SomeNonmodels........................................................... 30 ConsistencyandIndependence............................................... 32 SomeInfiniteModels....................................................... 33 ParallelPostulates.......................................................... 37 TheoremsinIncidenceGeometry ............................................ 42 Exercises.................................................................. 50 Chapter3. AxiomsforPlaneGeometry........................................ 53 PointsandLines............................................................ 54 Distance................................................................... 56 Segments.................................................................. 64 v vi Contents Rays...................................................................... 71 PlaneSeparation............................................................ 76 ConvexSets................................................................ 80 Exercises.................................................................. 81 Chapter4. Angles............................................................ 83 Angles..................................................................... 83 BetweennessofRays........................................................ 89 TheInteriorofanAngle..................................................... 94 Exercises.................................................................. 101 Chapter5. Triangles ......................................................... 103 Definitions.................................................................103 IntersectionsofLinesandTriangles.......................................... 105 CongruentTriangles........................................................ 106 Inequalities................................................................ 113 MoreCongruenceTheorems.................................................119 Exercises.................................................................. 122 Chapter6. ModelsofNeutralGeometry........................................123 TheCartesianModel........................................................124 ThePoincare´ DiskModel....................................................132 SomeNonmodels...........................................................135 IndependenceoftheNeutralGeometryPostulates..............................139 Exercises.................................................................. 140 Chapter7. PerpendicularandParallelLines.................................... 141 PerpendicularLines.........................................................141 ParallelLines .............................................................. 149 Exercises.................................................................. 154 Chapter8. Polygons ......................................................... 155 Polygons .................................................................. 155 ConvexPolygons........................................................... 157 NonconvexPolygons........................................................166 Exercises.................................................................. 174 Chapter9. Quadrilaterals..................................................... 175 ConvexQuadrilaterals.......................................................175 Parallelograms............................................................. 180 Exercises.................................................................. 183 Contents vii Chapter10. TheEuclideanParallelPostulate................................... 185 Angle-SumTheorems.......................................................189 QuadrilateralsinEuclideanGeometry........................................ 194 Exercises.................................................................. 196 Chapter11. Area............................................................ 199 PolygonalRegions..........................................................199 AdmissibleDecompositions................................................. 201 AreaFormulas............................................................. 204 IstheAreaPostulateIndependentoftheOthers?...............................209 Exercises.................................................................. 210 Chapter12. Similarity........................................................213 Definitions.................................................................213 TheSide-SplitterTheorem...................................................214 TriangleSimilarityTheorems................................................ 216 ProportionTheorems........................................................219 CollinearityandConcurrenceTheorems...................................... 221 TheGoldenRatio...........................................................224 PerimetersandAreasofSimilarFigures...................................... 226 Exercises.................................................................. 227 Chapter13. RightTriangles...................................................229 ThePythagoreanTheorem...................................................229 ApplicationsofthePythagoreanTheorem.....................................233 SimilarityRelationsinRightTriangles........................................236 Trigonometry .............................................................. 238 CategoricityoftheEuclideanPostulates.......................................243 Exercises.................................................................. 245 Chapter14. Circles.......................................................... 247 Definitions.................................................................247 CirclesandLines........................................................... 248 IntersectionsbetweenCircles................................................ 252 ArcsandInscribedAngles...................................................254 InscribedandCircumscribedPolygons........................................262 RegularPolygonsandCircles................................................271 Exercises.................................................................. 274 Chapter15. CircumferenceandCircularArea .................................. 279 Circumference ............................................................. 280 ApproximationsbyRegularPolygons.........................................285 viii Contents TheDefinitionofPi.........................................................288 AreaofaCircularRegion................................................... 290 Generalizations.............................................................293 Exercises.................................................................. 293 Chapter16. CompassandStraightedgeConstructions........................... 295 BasicConstructions.........................................................296 ConstructingRegularPolygons .............................................. 305 ImpossibleConstructions....................................................307 Exercises.................................................................. 319 Chapter17. TheParallelPostulateRevisited....................................321 PostulatesEquivalenttotheEuclideanParallelPostulate........................321 AngleSumsandDefects.................................................... 325 Clairaut’sPostulate......................................................... 330 It’sAllorNothing.......................................................... 333 Exercises.................................................................. 334 Chapter18. IntroductiontoHyperbolicGeometry...............................337 TheHyperbolicParallelPostulate............................................ 337 SaccheriandLambertQuadrilaterals..........................................340 AsymptoticRays........................................................... 343 AsymptoticTriangles....................................................... 350 Exercises.................................................................. 354 Chapter19. ParallelLinesinHyperbolicGeometry ............................. 355 ParallelLinesandAngles....................................................357 ParallelLinesandCommonPerpendiculars ................................... 359 DistancesbetweenParallelLines.............................................362 Exercises.................................................................. 367 Chapter20. Epilogue: WhereDoWeGofromHere?............................ 369 AppendixA. Hilbert’sAxioms................................................375 AppendixB. Birkhoff’sPostulates.............................................377 AppendixC. TheSMSGPostulates............................................379 AppendixD. ThePostulatesUsedinThisBook.................................381 PostulatesofNeutralGeometry.............................................. 381 PostulatesofEuclideanGeometry............................................382 PostulatesofHyperbolicGeometry...........................................382 Contents ix AppendixE. TheLanguageofMathematics.................................... 383 SimpleStatements..........................................................383 LogicalConnectives........................................................ 385 ConditionalStatements......................................................388 Quantifiers.................................................................394 MathematicalDefinitions....................................................401 Exercises.................................................................. 402 AppendixF. Proofs.......................................................... 405 TheStructureofMathematicalProofs ........................................ 405 DirectProofs...............................................................410 IndirectProofs............................................................. 416 ExistenceProofs............................................................417 ProofbyMathematicalInduction.............................................420 AppendixG. SetsandFunctions .............................................. 423 BasicConcepts.............................................................423 OrderedPairsandCartesianProducts.........................................426 Functions..................................................................428 Exercises.................................................................. 431 AppendixH. PropertiesoftheRealNumbers................................... 433 PrimitiveTerms............................................................ 433 Definitions.................................................................434 Properties..................................................................435 AppendixI. RigidMotions: AnotherApproach................................. 441 ReflectionImpliesSAS ..................................................... 442 SASImpliesReflection ..................................................... 446 Exercises.................................................................. 449 References....................................................................451 Index.........................................................................455