An excursion through elementary mathematics, Vol.2 PDF
Preview An excursion through elementary mathematics, Vol.2
Problem Books in Mathematics Antonio Caminha Muniz Neto An Excursion through Elementary Mathematics, Volume II Euclidean Geometry Problem Books in Mathematics SeriesEditor: PeterWinkler DepartmentofMathematics DartmouthCollege Hanover,NH USA Moreinformationaboutthisseriesathttp://www.springer.com/series/714 Antonio Caminha Muniz Neto An Excursion through Elementary Mathematics, Volume II Euclidean Geometry 123 AntonioCaminhaMunizNeto UniversidadeFederaldoCeará Fortaleza,Ceará,Brazil ISSN0941-3502 ISSN2197-8506 (electronic) ProblemBooksinMathematics ISBN978-3-319-77973-7 ISBN978-3-319-77974-4 (eBook) https://doi.org/10.1007/978-3-319-77974-4 LibraryofCongressControlNumber:2017933290 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart ofSpringerNature. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To mydearestwifeMonica, forallthatgoeswithoutsaying. Preface Thisisthesecondvolumeofaseriesofthreevolumes(theotheronesbeing [5]and [6]) devoted to the mathematics of mathematical olympiads. Generally speaking, theyaresomewhatexpandedversionsofacollectionofsixvolumes,firstpublished in Portuguese by the Brazilian Mathematical Society in 2012 and currently in its secondedition. The material collected here and in the other two volumes is based on course notesthatevolvedovertheyearssince1991,whenI firstbegancoachingstudents ofFortalezatotheBrazilianMathematicalOlympiadandtotheInternationalMath- ematical Olympiad. Some ten years ago, preliminary versions of the Portuguese texts also served as textbooksfor several editions of summer courses delivered at UFCtomathteachersoftheCapeVerdeRepublic. All volumes were carefully planned to be a balanced mixture of a smooth and self-containedintroductiontothe fascinatingworldofmathematicalcompetitions, aswellastoserveastextbooksforstudentsandinstructorsinvolvedwithmathclubs forgiftedhighschoolstudents. Upon writing the books, I have stuck myself to an invaluable advice of the eminent Hungarian-Americanmathematician George Pólya, who used to say that one cannot learn mathematics without getting one’s hands dirty. That’s why, in several points throughout the text, I left to the reader the task of checking minor aspectsofmoregeneraldevelopments.Theseappeareitherassmallomitteddetails in proofs or as subsidiary extensions of the theory. In this last case, I sometimes refer the reader to specific problemsalong the book, which are marked with an * and whose solutions are considered to be an essential part of the text. In general, in each section I collect a list of problems, carefully chosen in the direction of applyingthematerialandideaspresentedinthetext.Dozensofthemaretakenfrom formereditionsofmathematicalcompetitionsandrangefromthealmostimmediate to real challenging ones. Regardless of their level of difficulty, generoushints, or evencompletesolutions,areprovidedtovirtuallyallofthem. A quick glance through the Contents promptly shows that this second volume dealswith planeandsolid Euclideangeometry.Generallyspeaking,Chaps.1 to 9 vii viii Preface dealwithplanegeometry,whereastheremainingoneswithsolidgeometry.Wenow describethematerialcoveredabitmorespecifically. The text begins in a somewhat informal way, relying on the reader’s previous knowledgeofthebasicsofgeometryandemphasizingsimplegeometricconstruc- tions. Thisis donepurposefully,so that the axiomaticmethoddoesnotengulfthe exposition,fromthestart,withanamountofformalismunnecessaryforourgoals. Nevertheless, as the text evolves and deeper results are presented, the synthetic method of Euclid gains paramount importance, and from this time on several beautifulclassicaltheorems,usuallyabsentfromhighschooltextbooks,maketheir appearance. Afteraquickreviewofthemostelementaryconceptsandresults,Chaps.2to5 discussthecentralideasofcongruence,locus,similarity,andarea.Apartfromwhat isusuallyexpected,anumberofadditionaltopicsandresultsarediscussed,among whichPtolemy’sproblemonthe locusofpointswith prescribedratioofdistances totwoothergivenpoints,thecollinearityandconcurrenceresultsofMenelausand Ceva, some of Euler’s classical results on the geometryof the triangle, the notion of power of a point with respect to a circle and Apollonius’ tangency problems, and the isoperimetric problemfor triangles. Also, from a theoreticalstandpoint, a carefuldevelopmentofthenotionofareaandcircumferenceofacircleispresented inChap.5. The last four chapters dealing with plane geometry present analytic geometry, trigonometry,vectors,andsomeprojectivegeometryasdistinct,thoughinterrelated, tools for the study of plane Euclidean geometry. We do this without being too encyclopedic,soasnottoovershadowthecentralideas.Ontheonehand,webelieve this way we make it easier for the reader to grasp the role of each such portion of knowledge amid the whole of geometry. On the other hand, such additional methodsareappliedbothtoexpandthetheoryandtogetfurtherinsightonprevious resultsandexamples.Forinstance,thetextbringsthreesectionsonconics,twoin Chap.6usinganalyticandsynthetictools,andathirdoneinChap.10,usingsimple solidgeometryconceptstoextendtoconicsabunchofresultsofprojectivenature, discussedinChap.9. Reflecting the current trend in mathematical competitions, our exposition of solid geometry is shorter than that of plane geometry. Nevertheless, it covers all of what is usually present in high school curricula, as well as some other more profound topics. Among these, we would like to mention the representation of conics as conic sections, the use of central projections to the study of some projective properties of conics, the discussion of some aspects of the interesting classofisoscelestetrahedra,thepresentationofacomplete—thoughsimple—proof of Euler’s theorem on convex polyhedra,the classification and construction of all regular polyhedra,as well as the computationof their volumes, and a glimpse on inversioninthree-dimensionalspace.Sincethereaderisexpectedtoreachthesolid geometry chapters with a thorough grounding on plane geometry, some of these topicsarepartiallycoveredamidtheproposedproblems.However,wheneverwedo so,weprovideessentiallyfullsolutionstothem. Preface ix Several people and institutions contributed throughoutthe years for my efforts of turning a bunch of handwritten notes into these books. The State of Ceará Mathematical Olympiad, created by the Mathematics Department of the Federal University of Ceará (UFC) back in 1980 and now in its 37th edition, has since then motivated hundreds of youngsters of Fortaleza to deepen their studies of mathematics.Iwasonesuchstudentinthelate1980s,andmyinvolvementwiththis competitionandwiththeBrazilianMathematicalOlympiadafewyearslaterhada decisive influenceon my choice of career. Throughoutthe 1990s,I had the honor of coaching several brilliant students of Fortaleza to the Brazilian Mathematical Olympiad.SomeofthementeredBrazilianteamstotheIMOorotherinternational competitions, and their doubts, comments, and criticisms were of great help in shaping my view on mathematical competitions. In this sense, sincere thanks go toJoãoLuizdeA.A.Falcão,RoneyRodgerS.deCastro,MarceloM.deOliveira, MarcondesC.FrançaJr.,MarceloC.deSouza,EduardoC.Balreira,BrenodeA.A. Falcão,FabrícioS.Benevides,RuiF.Vigelis,DanielP.Sobreira,SamuelB.Feitosa, DaviMáximoA.Nogueira,andYuriG.Lima. Professor João Lucas Barbosa, upon inviting me to write the textbooks to the Amílcar Cabral Educational Cooperation Project with Cape Verde Republic, had unconsciouslyprovidedmewiththemotivationtocompletethePortugueseversion of these books. The continuoussupportof Professor Hilário Alencar, presidentof theBrazilianMathematicalSocietywhenthePortugueseeditionwasfirstpublished, was also of great importance for me. Special thanks go to my colleagues— Professors Samuel B. Feitosa and Fernanda E. C. Camargo—who read the entire English version and helped me improve it in a number of ways. If it weren’t for myeditoratSpringer-Verlag,Mr.RobinsondosSantos,Ialmostsurelywouldnot havehadthecouragetoembracethetaskoftranslatingmorethan1500pagesfrom PortugueseintoEnglish.IacknowledgeallthestaffofSpringerinvolvedwiththis projectinhisname. Finally,andmostly,Iwouldlike toexpressmydeepestgratitudetomyparents Antonioand Rosemary,my wife Monica, and our kidsGabriel and Isabela. From early childhood,my parentshave always called my attention to the importanceof asolideducation,havingdonealltheycouldformeandmybrotherstoattendthe best possible schools. My wife and kids fulfilled our home with the harmonyand softnessIneededtogettoendureonseveralmonthsofworkwhiletranslatingthis book. Fortaleza,Brazil AntonioCaminhaMunizNeto December2017 Contents 1 BasicGeometricConcepts................................................. 1 2 CongruenceofTriangles................................................... 19 3 LociinthePlane............................................................ 61 4 ProportionalityandSimilarity............................................ 101 5 AreaofPlaneFigures ...................................................... 151 6 TheCartesianMethod ..................................................... 181 7 TrigonometryandGeometry.............................................. 227 8 VectorsinthePlane ........................................................ 269 9 AFirstGlimpseonProjectiveTechniques............................... 289 10 BasicConceptsinSolidGeometry........................................ 331 11 SomeSimpleSolids......................................................... 377 12 ConvexPolyhedra .......................................................... 403 13 VolumeofSolids ............................................................ 427 14 HintsandSolutions......................................................... 449 Glossary........................................................................... 535 Bibliography...................................................................... 537 Index............................................................................... 539 xi
The list of books you might like
