Foundations of geometry for university students an

arXiv:math/0702029v1[math.HO]1Feb2007MINISTRYOFGENERALANDPROFESSIONALEDUCATIONOFRUSSIANFEDERATIONBASHKIRSTATEUNIVERSITYSHARIPOVR.A.FOUNDATIONSOFGEOMETRYFORUNIVERSITYSTUDENTSANDHIGHSCHOOLSTUDENTSThetextbookUfa19982UDC541.1SharipovR.A.Foundationsofgeometryforuniversitystudentsandhigh-schoolstudents.Thetextbook—Ufa,1998.—220pages—ISBN5-7477-02494-1.Thisbookisatextbookforthecourseoffoundationsofgeometry.ItisaddressedtomathematicsstudentsinUniversitiesandtoHighSchoolstudentsfordeeperlearningtheelementarygeometry.Itcanalsobeusedinmathematicscoteriesandself-educationgroups.InpreparingRussianeditionofthisbookIusedcomputertypesettingonthebaseofAMS-TEXpackageandIusedCyrillicfontsofLh-familydistributedbyCyrTUGassociationofCyrillicTEXusers.EnglisheditionisalsotypesetbyAMS-TEX.Referees:Prof.R.R.Gadylshin,BashkirStatePedagogicalUniversity(BGPU);Prof.E.M.Bronshtein,UfaStateUniversityforAircraftandTechnology(UGATU);Mrs.N.V.Medvedeva,HonoredteacheroftheRepublicBashkortostan,HighSchoolNo.42,Ufa.ISBN5-7477-02494-1cSharipovR.A.,1998EnglishTranslationcSharipovR.A.,20073CONTENTS.CONTENTS.......................................................................3.PREFACE..........................................................................6.CHAPTERI.EUCLID’SGEOMETRY.ELEMENTSOFTHESETTHEORYANDAXIOMATICS...............7.§1.Someinitialconceptsofthesettheory...........................7.§2.Equivalencerelationsandbreakingintoequivalenceclasses........................................................9.§3.Orderedsets...............................................................10.§4.Ternaryrelations........................................................11.§5.Settheoreticterminologyingeometry..........................11.§6.Euclid’saxiomatics.....................................................12.§7.Setsandmappings......................................................12.§8.Restrictionandextensionofmappings.........................16.CHAPTERII.AXIOMSOFINCIDENCEANDAXIOMSOFORDER......................................18.§1.Axiomsofincidence....................................................18.§2.Axiomsoforder..........................................................24.§3.Segmentsonastraightline........................................30.§4.Directions.Vectorsonastraightline...........................34.§5.Partitioningastraightlineandaplain.........................40.§6.Partitioningthespace.................................................45.CHAPTERIII.AXIOMSOFCONGRUENCE.................49.§1.Binaryrelationsofcongruence.....................................49.§2.Congruenceofsegments..............................................49.§3.Congruenttranslationofstraightlines........................56.4§4.Slippingvectors.Additionofvectorsonastraightline........................................................59.§5.Congruenceofangles..................................................64.§6.Arightangleandorthogonality...................................73.§7.Bisectionofsegmentsandangles..................................78.§8.Intersectionoftwostraightlinesbyathirdone...........81.CHAPTERIV.CONGRUENTTRANSLATIONSANDMOTIONS.......................................................84.§1.Orthogonalityofastraightlineandaplane..................84.§2.Aperpendicularbisectorofasegmentandtheplaneofperpendicularbisectors.............................88.§3.Orthogonalityoftwoplanes.......................................89.§4.Adihedralangle.........................................................93.§5.Congruenttranslationsofaplaneandthespace...........97.§6.Mirrorreflectioninaplaneandinastraightline........105.§7.Rotationofaplaneaboutapoint..............................106.§8.Thetotalrotationgroupandthegroupofpurerotationsofaplane..................................................111.§9.Rotationofthespaceaboutastraightline.................113.§10.Thetheoremonthedecompositionofrotations.........116.§11.Thetotalrotationgroupandthegroupofpurerotationsofthespace..............................................121.§12.Orthogonalprojectionontoastraightline................122.§13.Orthogonalprojectionontoaplane..........................124.§14.Translationbyavectoralongastraightline.............129.§15.Motionsandcongruenceofcomplicatedgeometricforms......................................................134.CHAPTERV.AXIOMSOFCONTINUITY...................138.§1.Comparisonofstraightlinesegments.........................138.§2.Comparisonofangles................................................141.§3.Axiomsofrealnumbers.............................................146.§4.Binaryrationalapproximationsofrealnumbers..........150.5§5.TheArchimedesaxiomandCantor’saxiomingeometry..............................................................154.§6.Therealaxis............................................................156.§7.Measuringstraightlinesegments...............................161.§8.Similaritymappingsforstraightlines.Multiplicationofvectorsbyanumber.............................................167.§9.Measu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