AMS / MAA TEXTBOOKS VOL 44 Geometry: The Line and the Circle VOL AMS / MAA TEXTBOOKS 44 Maureen T. Carroll and Elyn Rykken Geometry: The Line and the Circle is an undergraduate text with a strong G narrative that is written at the appropriate level of rigor for an upper-level e survey or axiomatic course in geometry. Starting with Euclid’s Elements, o the book connects topics in Euclidean and non-Euclidean geometry in an m intentional and meaningful way, with historical context. e The line and the circle are the principal characters driving the narrative. t In every geometry considered—which include spherical, hyperbolic, and M r taxicab, as well as finite affine and projective geometries—these two a y u objects are analyzed and highlighted. Along the way, the reader contem- re : T plates fundamental questions such as: What is a straight line? What does e n h parallel mean? What is distance? What is area? T . e C There is a strong focus on axiomatic structures throughout the text. While a Euclid is a constant inspiration and the Elements is repeatedly revisited rr L o with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean ll in a geometries are introduced very early to give the reader perspective n e d on questions of axiomatics. Rounding out the thorough coverage of E axiomatics are concluding chapters on transformations and constructa- ly a n n bility. The book is compulsively readable with great attention paid to the R historical narrative and hundreds of attractive problems. y d k k e t n h e C i r c l e For additional information and updates on this book, visit www.ams.org/bookpages/text-44 A M S / M A A P TEXT/44 R E S S 4-Color Process 496 pages on 50lb stock • Backspace 1 11/16'' Geometry: The Line and the Circle AMS/MAA TEXTBOOKS VOL 44 Geometry: The Line and the Circle Maureen T. Carroll Elyn Rykken CommitteeonBooks JenniferJ.Quinn,Chair MAATextbooksEditorialBoard StanleyE.Seltzer,Editor BelaBajnok SuzanneLynneLarson JeffreyL.Stuart MatthiasBeck JohnLorch RonD.Taylor,Jr. HeatherAnnDye MichaelJ.McAsey ElizabethThoren WilliamRobertGreen VirginiaNoonburg RuthVanderpool CharlesR.Hampton 2010MathematicsSubjectClassification.Primary51-01. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/text-44 LibraryofCongressCataloging-in-PublicationData Names:Carroll,MaureenT.,1966–author.|Rykken,Elyn,1967–author. Title:Geometry:Thelineandthecircle/MaureenT.Carroll,ElynRykken. Description:Providence,RhodeIsland:MAAPress,animprintoftheAmericanMathematicalSociety, [2018]|Series:AMS/MAAtextbooks;volume44|Designedforanupper-levelcollegegeometrycourse. |Includesbibliographicalreferencesandindexes. Identifiers:LCCN2018034790|ISBN9781470448431(alk.paper) Subjects:LCSH:Geometry–Textbooks.|Geometry–Studyandteaching(Higher) Classification:LCCQA445.C29852018|DDC516–dc23 LCrecordavailableathttps://lccn.loc.gov/2018034790 Copyingandreprinting. Individualreadersofthispublication,andnonprofitlibrariesactingforthem, arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,providedthecustomaryac- knowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationispermit- tedonlyunderlicensefromtheAmericanMathematicalSociety.Requestsforpermissiontoreuseportions ofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. Formoreinformation,please visitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. ©2018bytheauthors.Allrightsreserved. PrintedintheUnitedStatesofAmerica. ⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 232221201918 Toourparentsfortheirlove,supportandencouragement Contents NotetotheInstructor xi Outlineofthebook xii Designingacourseusingthistext xiii NotetotheReader xvii Acknowledgments xix 1 TheLineandtheCircle 1 1.1 Introduction 1 1.2 Whichcamefirst? 2 1.3 Whatisastraightline,anyways? 4 2 Euclid’sElements: DefinitionsandAxioms 7 2.1 TheElements 7 2.2 Definitions 9 2.3 Postulatesandcommonnotions 12 3 BookIofEuclid’sElements: NeutralGeometry 17 3.1 PropositionsI.1throughI.8 18 3.2 PropositionsI.9throughI.15 32 3.3 PropositionsI.16throughI.28andI.31 41 4 SphericalGeometry 59 4.1 Whatisastraightline,anyways? -Part2 60 4.2 TrianglesinSphericalgeometry 65 4.3 Euclid’saxiomsviewedinSphericalgeometry 68 4.4 Neutralgeometryonthesphere 71 4.5 AreainSphericalgeometry 77 4.6 Trigonometryforsphericaltriangles 82 4.7 Uniquelysphericalconstructions 88 5 TaxicabGeometry 93 5.1 Points,lines,angles,distancesandcircles 94 5.2 Euclid’spostulatesinTaxicabgeometry 100 5.3 CongruenceschemesinTaxicabgeometry 101 5.4 TherestofNeutralgeometry 102 6 HilbertandGödel 105 6.1 Axiomaticsystems 106 6.2 AFourPointgeometry 110 vii viii Contents 6.3 Hilbert’saxiomsforEuclideanplanegeometry 113 6.4 SphericalandTaxicabgeometries 120 6.5 Gödelandconsistency 123 7 BookI:Non-NeutralGeometry 127 7.1 Parallellines 128 7.2 PropositionsI.32andI.33 132 7.3 Area 135 7.4 PropositionsI.34throughI.41 138 7.5 PropositionsI.42throughI.46 145 7.6 ThePythagoreanTheorem 151 8 BookII:GeometricAlgebra 159 8.1 PropositionII.1throughII.10 160 8.2 PropositionsII.11throughII.14 165 8.3 Quadratureonthesphere 171 9 BookVI:Similarity 175 9.1 BookV:Ratioandproportion 176 9.2 Similarity 176 9.3 AgeneralizedPythagoreanTheorem 184 10 BookIII:Circles 189 10.1 Definitions 190 10.2 Tangency 193 10.3 Arcs,chordsandangles 201 10.4 AreaPropositions: III.35throughIII.37 210 10.5 Thecircumferenceofacircle&𝜋 216 11 BookIV:Circles&Polygons 221 11.1 Definitions 222 11.2 Circles&triangles 223 11.3 Circles&squares 236 11.4 Circles&pentagons 241 11.5 Constructingregularpolygons 246 11.6 Theareaofacircle&𝜋 251 12 ModelsfortheHyperbolicPlane 261 12.1 Historicaloverview 262 12.2 Modelsofthehyperbolicplane 266 12.3 Arclength&distanceinthehalf-plane model 277 13 AxiomaticHyperbolicGeometry 289 13.1 Parallellines 291 13.2 Omegatriangles 302 13.3 Saccheriquadrilaterals 307 13.4 Hyperbolicarea 320 Contents ix 14 FiniteGeometries 331 14.1 FourPointgeometry-Part2 332 14.2 Fano’splane 335 14.3 Projectivegeometry 339 14.4 Affineplanes 355 14.5 Transformingafineintoprojective 366 14.6 Openprobleminfinitegeometry 370 15 Isometries 373 15.1 Rigidmotionsorisometries 374 15.2 Reflections 378 15.3 IsometriesoftheEuclideanplane 382 15.4 InversionsintheEuclideanplane 399 15.5 Isometriesofthehyperbolicplane 406 16 Constructibility 415 16.1 Fourfamousproblemsofantiquity 416 16.2 Constructiblenumbers 419 16.3 Fourcounterexamples 429 16.4 Thelimitsofgeometry 435 AppendixA Euclid’sDefinitionsandAxioms 437 A.1 Definitions 437 A.2 Postulates 438 A.3 Commonnotions 438 AppendixB Euclid’sPropositions 439 B.1 BookI 439 B.2 BookII 442 B.3 BookIII 443 B.4 BookIV 444 B.5 BookVI 445 AppendixC VisualGuidetoEuclid’sPropositions 447 C.1 BookI 447 C.2 BookII 453 C.3 BookIII 454 C.4 BookIV 456 AppendixD Euclid’sProofs 457 D.1 BookI 457 AppendixE Hilbert’sAxiomsforPlaneEuclideanGeometry 461 Credits,PermissionsandAcknowledgements 463 Bibliography 465 NotationIndex 471 Index 473