Table Of ContentProblem 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
