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Mathematics via Problems PART 2: Geometry Alexey A. Zaslavsky & Mikhail B. Skopenkov
Mathematics via Problems PART 2: Geometry Mathematical Circles Library
Mathematics via Problems PART 2: Geometry Alexey A. Zaslavsky & Mikhail B. Skopenkov
Translated from Russian by Paul Zeitz and Sergei G. Shubin
Berkeley,CaliforniaProvidence,RhodeIsland
AdvisoryBoardfortheMSRI/MathematicalCirclesLibrary
TituAndreescuZvezdelinaStankova
DavidAucklyJamesTanton
H´el`eneBarceloRaviVakil
ZumingFengDianaWhite
TonyGardinerIvanYashchenko
AndyLiuPaulZeitz
AlexanderShenJoshuaZucker
TatianaShubin(Chair)
ScientificEditor:DavidScott
ThisworkwasoriginallypublishedinRussianby“MCNMO”underthetitle pementymatematikivzadaqah, c 2018.Thepresenttranslationwascreated underlicensefortheAmericanMathematicalSocietyandispublishedbypermission.
ThisvolumeispublishedwiththegeneroussupportoftheSimons FoundationandTomLeightonandBonnieBergerLeighton.
2020 MathematicsSubjectClassification.Primary00-01, 00A07,51-01,52-01,14-01,97G10,97-01.
Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/mcl-26
LibraryofCongressCataloging-in-PublicationData
Names:Zaslavski˘ı,Alekse˘ıAleksandrovich,1960–author. | Skopenkov,MikhailB.,1983–author.
Title:Mathematicsviaproblems.Part2.Geometry/AlexeyA.Zaslavsky,MikhailB.Skopenkov; translatedbySergeiShubinandPaulZeitz.
Othertitles: ˙ Elementymatematikivzadachakh.English
Description:Berkeley,California:MSRIMathematicalSciencesResearchInstitute;Providence, RhodeIslandAmericanMathematicalSociety,[2021] | Series:MSRImathematicalcircles library,1944-8074;26 | Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020057238 | ISBN9781470448790(paperback) | 9781470465216(ebook)
Subjects:LCSH:Geometry–Studyandteaching. | Problemsolving. | Geometry–Problems,exercises,etc. | AMS:General–Instructionalexposition(textbooks,tutorialpapers,etc.). | General–Generalandmiscellaneousspecifictopics–Problembooks. | Geometry–Instructional exposition(textbooks,tutorialpapers,etc.). | Convexanddiscretegeometry–Instructional exposition(textbooks,tutorialpapers,etc.). | Algebraicgeometry–Instructionalexposition (textbooks,tutorialpapers,etc.). | Mathematicseducation–Geometry–Comprehensive works. | Mathematicseducation–Instructionalexposition(textbooks,tutorialpapers,etc.). Classification:LCCQA462.Z372021 | DDC516.0071–dc23 LCrecordavailableathttps://lccn.loc.gov/2020057238
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∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurep ermanenceanddurability. VisittheAMShomepageat https://www.ams.org/ 10987654321262524232221
Contents Forewordix Problems,exercises,circles,andolympiadsix Whythisbook,andhowtouseitx English-languagereferencesxi Introductionxiii Whatthisbookisaboutandwhoitisforxiii Learningbysolvingproblemsxiv Partingwords ByA.Ya.Kanel-Belov xv Olympiadsandmathematicsxv Researchproblemsforhighschoolstudentsxvi Howthisbookisorganizedxvi Resourcesandliteraturexvi Acknowledgmentsxvii Numberingandnotationxvii Notationxviii Referencesxix
Chapter1.Triangle1
1.Carnot’sprinciple(1) ByV.Yu.ProtasovandA.A.Gavrilyuk 2 Suggestions,solutions,andanswers3
2.Thecenteroftheinscribedcircle(2) ByV.Yu.Protasov 4 Suggestions,solutions,andanswers5
3.TheEulerline ByV.Yu.Protasov 7 Suggestions,solutions,andanswers7
4.Carnot’sformula(2∗ ) ByA.D.Blinkov 8 Suggestions,solutions,andanswers9
5.Theorthocenter,orthotriangle,andnine-pointcircle(2) ByV.Yu.Protasov 11 Suggestions,solutions,andanswers13
6.Inequalitiesinvolvingtriangles(3∗ ) ByV.Yu.Protasov 13 Suggestions,solutions,andanswers14
7.Bisectors,heights,andcircumcircles(2) ByP.A.Kozhevnikov 15 Suggestions,solutions,andanswers16 v
8.“Semi-inscribed”circle(3∗ ) ByP.A.Kozhevnikov 19 Mainseriesofproblems—119 Mainseriesofproblems—220 Supplementaryproblems—120 Supplementaryproblems—220 Suggestions,solutions,andanswers21
9.ThegeneralizedNapoleon’stheorem(2∗ )
ByP.A.Kozhevnikov 25 Introductoryproblems25 FormulationandproofofthegeneralizedNapoleon’stheorem26 Suggestions,solutions,andanswers27
10.IsogonalconjugationandtheSimsonline(3∗ ) ByA.V.Akopyan 31 Suggestions,solutions,andanswers33 Additionalreading38
Chapter2.Circle39
1.Thesimplestpropertiesofacircle(1) ByA.D.Blinkov 39 Suggestions,solutions,andanswers40
2.Inscribedangles(1) ByA.D.BlinkovandD.A.Permyakov 42 Suggestions,solutions,andanswers44
3.Inscribedandcircumscribedcircles(2) ByA.A.Gavrilyuk 46 Suggestions,solutions,andanswers47
4.Theradicalaxis(2) ByI.N.ShnurnikovandA.I.Zasorin
5.Tangency(2) ByI.N.ShnurnikovandA.I.Zasorin
6.Ptolemy’sandCasey’sTheorems(3∗ ) ByA.D.Blinkovand A.A.Zaslavsky
6.A.Ptolemy’sTheorem50 6.B.Casey’sTheorem51 Suggestions,solutions,andanswers52
Chapter3.Geometrictransformations55
1.Applicationsoftransformations(1) ByA.D.Blinkov 55 Suggestions,solutions,andanswers57
2.Classificationofisometriesoftheplane(2) ByA.B.Skopenkov 61 Hints63
3.Classificationofisometriesofspace(3*) ByA.B.Skopenkov 63 Hints65
4.Anapplicationofsimilarityandhomothety(1) ByA.D.Blinkov 65 Suggestions,solutions,andanswers67
5.Rotationalhomothety(2) ByP.A.Kozhevnikov 71
5.A.Introductoryproblemsinvolvingcyclists71
5.B.Mainproblems72
5.C.Additionalproblems73
Suggestions,solutions,andanswers73
6.Similarity(1) ByA.B.Skopenkov 76
7.Dilationtoaline(2) ByA.Ya.Kanel-Belov 77 Suggestions,solutions,andanswers78
8.Parallelprojectionandaffinetransformations(2) ByA.B.Skopenkov 78
Suggestions,solutions,andanswers80
9.Centralprojectionandprojectivetransformations(3) ByA.B.Skopenkov 81
10.Inversion(2) ByA.B.Skopenkov 83 Additionalreading86
Chapter4.Affineandprojectivegeometry87
1.Masspoints(2) ByA.A.Gavrilyuk 87 Suggestions,solutions,andanswers89
2.Thecross-ratio(2) ByA.A.Gavrilyuk 90 Suggestions,solutions,andanswers92
3.Polarity(2) ByA.A.GavrilyukandP.A.Kozhevnikov 93 Fundamentalpropertiesandintroductoryproblems93 Mainproblems94 Additionalproblems95 Suggestions,solutions,andanswers96 Additionalreading98
Chapter5.Complexnumbersandgeometry(3) ByA.A.Zaslavsky 99 1.Complexnumbersandelementarygeometry99 Suggestions,solutions,andanswers101 2.ComplexnumbersandMöbiustransformations102 Additionalproblems103 Suggestions,solutions,andanswers103 Additionalreading104
Chapter6.Constructionsandloci105
1.Loci(1) ByA.D.Blinkov 105 Suggestions,solutions,andanswers106 2.Constructionandlociproblemsinvolvingarea(1) ByA.D.Blinkov 111 Suggestions,solutions,andanswers113 3.Constructiontoolbox(2) ByA.A.Gavrilyuk 117 Suggestions,solutions,andanswers119 4.Auxiliaryconstructions(2∗ ) ByI.I.Shnurnikov 120 Suggestions,solutions,andanswers121 Additionalreading123
Chapter7.Solidgeometry125
1.Drawing(2) ByA.B.Skopenkov 125
viiiCONTENTS
Suggestions,solutions,andanswers126
2.Projections(2) ByM.A.Korchemkina 126
2.A.Projectionsoffiguresconstructedfromcubes126
2.B.Trajectories128
3.Regularpolyhedra(3)130
3.A.Inscribedandcircumscribedpolyhedra
ByA.Ya.Kanel-Belov 130
Suggestions,solutions,andanswers132
3.B.Symmetries
ByA.B.Skopenkov
4.Higher-dimensionalspace(4∗ ) ByA.Ya.Kanel-Belov
4.A.Simplestpolyhedrainhigher-dimensionalspace
ByYu.M.BurmanandA.Ya.Kanel-Belov 134
4.B.Multi-dimensionalvolumes138
4.C.Volumesandintersections139
4.D.Researchproblems140
4.E.Partitionsintopartsofsmallerdiameter
ByA.M.Raigorodsky 141
Suggestions,solutions,andanswers141 Additionalreading143
Chapter8.Miscellaneousgeometryproblems145
1.Geometricoptimizationproblems(2) ByA.D.Blinkov 145
Suggestions,solutions,andanswers147
2.Area(2) ByA.D.Blinkov 151
Suggestions,solutions,andanswers152
3.Conicsections(3∗ ) ByA.V.Akopyan 158
Suggestions,solutions,andanswers161 4.Curvilineartrianglesandnon-Euclideangeometry(3∗ ) ByM.B.Skopenkov 166
Additionalproblems168
Suggestions,solutions,andanswers169 Additionalreading169
Bibliography171 Index175
Foreword Problems,exercises,circles,andolympiads
ThisisatranslationofPart2ofthebook MathematicsviaProblems by A.B.Skopenkov,M.B.Skopenkov,andA.A.Zaslavsky,andispartofthe AMS/MSRIMathematicalCirclesLibraryseries.Thegoalofthisseriesis tobuildabodyofworksinEnglishthathelptospreadthe“mathcircle” culture.
A mathematicalcircle isaneasternEuropeannotion.Mathcirclesare similartowhatmostAmericanswouldcallamathclubforkids,butwith severalimportantdistinguishingfeatures.
First,theyare verticallyintegrated :youngstudentsmayinteractwith olderstudents,collegestudents,graduatestudents,industrialmathematicians,professors,andevenworld-classresearchers,allinthesameroom. Thecircleisnotsomuchaclassroomasagatheringofyounginitiateswith eldertribespeople,whopassdown folklore.
Second,the“curriculum,”suchasitis,isdominatedby problems rather thanspecificmathematicaltopics.Aproblem,incontrasttoanexercise, isamathematicalquestionthatonedoesn’tknowhow,atleastinitially,to approach.Forexample,“Whatis3times5?”isanexerciseformostpeople butaproblemforaveryyoungchild.Computing 534 isalsoanexercise, conceptuallyverymuchlikethefirstexample,certainlyharder,butonlyin a“technical”sense.Andaquestionlike“Evaluate 7 2 e5x sin3xdx”isalso anexercise—forcalculusstudents—amatterof“merely”knowingtheright algorithmandhowtoapplyit.
Problems,bycontrast,donotcomewithalgorithmsattached.Bytheir verynature,theyrequire investigation,whichisbothanartandascience, demandingtechnicalskillalongwithfocus,tenacity,andinventiveness.Math circlesteachstudentstheseskills,notwithformalinstruction,butbyhaving them domath andobserveothersdoingmath.Studentslearnthataproblem worthsolvingmayrequirenotminutesbutpossiblyhours,days,orevenyears ofeffort.Theyworkonsomeoftheclassicfolkloreproblemsanddiscover howtheseproblemscanhelptheminvestigateotherproblems.Theylearn hownottogiveupandhowtoturnerrorsorfailuresintoopportunitiesfor moreinvestigation.Achildinamathcirclelearnstodoexactlywhata
xFOREWORD researchmathematiciandoes;indeed,heorshedoesindependentresearch, albeitatalowerlevel,andoften—althoughnotalways—onproblemsthat othershavealreadysolved.
Finally,manymathcircleshaveaculturesimilartothatofasports team,withintensecamaraderie,respectforthe“coach,”andhealthycompetitiveness(managedwisely,ideally,bytheleader/facilitator).Themath circlecultureisoftencomplementedbyavarietyofproblemsolvingcontests, oftencalled olympiads.Amathematicalolympiadproblemis,firstofall,a genuineproblem(atleastforthecontestant),andusuallyrequiresananswer whichis,ideally,awell-writtenargument(a“proof”).
Whythisbook,andhowtouseit TheMathematicalCirclesLibraryeditorialboardchosetotranslatethis workfromRussianintoEnglishbecausethisbookhasanaudaciousgoal— promisedbyitstitle—todevelopmathematicsthroughproblems.Thisis notanoriginalidea,norjustaRussianone.AmericanuniversitieshaveexperimentedforyearswithIBL(inquiry-basedlearning)andMoore-method courses,structuredmethodsforteachingadvancedmathematicsthrough open-endedproblemsolving.1
Buttheauthors’massiveworkisanattempttocuratesequencesofproblemsforsecondarystudents(thestatedfocusisonhighschoolstudents,but thatcanbebroadlyinterpreted)thatallowthemtodiscoverandrecreate muchof“elementary”mathematics(numbertheory,polynomials,inequalities,calculus,geometry,combinatorics,gametheory,probability)andstart edgingintothesophisticatedworldofgrouptheory,Galoistheory,etc.
Thebookisimpossibletoreadfromcovertocover—norshoulditbe. Instead,thereaderisinvitedtostartworkingonproblemsthatheorshe findsappealingandchallenging.Manyoftheproblemshavehintsandsolutionsketches,butnotall.Noreaderwillsolvealltheproblems.That’s notthepoint—itisnotacontest.Furthermore,someoftheproblemsare notsupposedtobesolved,butshouldratherbepondered.Forexample,as soonasitis“technically”possibletosolveit,thebookintroducesApollonius’sproblem(Problem3.10.6inChapter3),oneofthedeepestandmost famouschallengesofclassicalgeometry,andthetextprovidesreferencesfor learningmoreaboutit.Justbecauseitis“tooadvanced”doesn’tmeanthat itshouldn’tbethoughtabout!
Indeed,thisisthephilosophyofthebook:mathematicsisnotasequentialdiscipline,whereoneispresentedwithadefinitionthatleadstoalemma whichleadstoatheoremwhichleadstoaproof.Insteaditisanadventure, filledwithexcitingsidetripsaswellaswildgoosechases.Theadventure isitsownreward,butitalso,fortuitously,leadstoadeepunderstanding
1 See,forexample,https://en.wikipedia.org/wiki/Moore_methodandhttp://www. jiblm.org.
andappreciationofmathematicalideasthatcannotbeachievedbypassive reading.
English-languagereferences
MostofthereferencescitedinthisbookareinRussian.However,thereare afewexcellentbooksinEnglish.Ourtwofavoritesaretheclassic Geometry Revisited [CG67]and[Che16],arecentandverycomprehensiveguideto “olympiadgeometry.”
PaulZeitz June2020
Introduction Whatthisbookisaboutandwhoitisfor
Adeepunderstandingofmathematicsisusefulbothformathematiciansand forhigh-techprofessionals.Inparticular,the“profession”inthetitleofthis bookdoesnotnecessarilymeantheprofessionofmathematics.
Thisbookisintendedforhighschoolstudentsandundergraduates(in particular,thoseinterestedinolympiads).Formoredetails,see“Olympiads andmathematics”onp.xv.Thebookcanbeusedbothforself-studyand forteaching.
Thisbookattemptstobuildabridge(byshowingthatthereisnogap) betweenordinaryhighschoolexercisesandthemoresophisticated,intricate, andabstractconceptsinmathematics.Thefocusisonengagingawideaudienceofstudentstothinkcreativelyinapplyingtechniquesandstrategies toproblemsmotivatedby“realworldorrealwork”[Mey].Studentsareencouragedtoexpresstheirideas,conjectures,andconclusionsinwriting.Our goalistohelpstudentsdevelopahostofnewmathematicaltoolsandstrategiesthatwillbeusefulbeyondtheclassroomandinanumberofdisciplines [IBL, Mey, RMP].
Thebookcontainsthemoststandard“base”material(althoughweexpectthatatleastsomeofthismaterialwillbereview—thatnotallisbeing learnedforthefirsttime).Butthemaincontentofthebookismorecomplex material.Sometopicsarenotwellknowninthetraditionsofmathematical circles,butareusefulbothformathematicaleducationandforpreparation forolympiads.
Thebookisbasedonclassestaughtbytheauthorsatdifferenttimesat theIndependentUniversityofMoscow,attheNationalResearchUniversity HigherSchoolofEconomics,atvariousMoscowschools,inpreparingthe RussianteamfortheInternationalMathematicalOlympiad,inthe“ContemporaryMathematics”summerschool,intheKirovandKostromaSummerMathematicalSchools,intheMoscowvisitingOlympiadSchool,inthe “MathematicalSeminar”and“OlympiadandMathematics”circles,andalso atthesummerConferenceoftheTournamentofTowns.
Muchofthisbookisaccessibletohighschoolstudentswithastrong interestinmathematics.2 Weprovidedefinitionsofconceptsthatarenot standardintheschoolcurriculum,orprovidereferences.However,many topicsaredifficultifyoustudythem“fromscratch.”Thus,theorderingof theproblemshelpstoprovide“scaffolding.”Atthesametime,manytopics are independent ofeachother.Formoredetails,seep.xvi,“Howthisbook isorganized”.
Learningbysolvingproblems Wesubscribetothetraditionofstudyingmathematicsbysolvinganddiscussingproblems.Theseproblemsareselectedsothatintheprocessof solvingthemthereader(moreprecisely,thesolver)mastersthefundamentalsofimportantideas,bothclassicalandmodern.Themainideasare developedincrementallywitholympiad-styleexamples—inotherwords,by thesimplestspecialcases,freefromtechnicaldetails.Inthisway,weshow howyoucanexploreanddiscovertheseideasonyourown.
Learningbysolvingproblemsisnotjustaseriousapproachtomathematicsbutalsocontinuesavenerableculturaltradition.Forexample,thenovices inZenmonasteriesstudybyreflectingonriddles(“koans”)giventothemby theirmentors.(However,theseriddlesarerathermorelikeparadoxesthan whatweconsidertobeproblems.)See,forexample,[Su];comparewith [Pl,pp.26–33].“Math”examplescanbefoundin[Ar04, BSh, GDI, KK08, Pr07-1, PoSe, SCY, Sk09, Va87-1, Zv],whichsometimesdescribenotonly problemsbutalsotheprinciplesofselectingappropriateproblems.Forthe Americantradition,see[IBL, Mey, RMP].
Learningbysolvingproblemsisdifficult,inpart,becauseitgenerally doesnotcreatethe illusion ofunderstanding.However,one’seffortsarefully rewardedbyadeepunderstandingofthematerial,atfirst,withtheability tocarryoutsimilar(andsometimesratherdifferent)reasoning.Eventually, whileworkingonfascinatingproblems,readerswillbefollowingthethought processesofthegreatmathematiciansandmayseehowimportantconcepts andtheoriesnaturallyevolve.Hopefullythiswillhelpthemmaketheirown equallyusefuldiscoveries(notnecessarilyinmath)!
Solvingaproblem,theoretically,requiresonlyunderstandingitsstatement.Otherfactsandconceptsarenotneeded.(However,usefulfactsand ideaswillbedevelopedwhensolvingselectedproblems.)Andyoumayneed toknowthingsfromotherpartsofthebook,asindicatedintheinstructions andhints.Forthemostimportantproblemsweprovidehints,instructions, solutions,andanswers,locatedattheendofeachsection.However,these shouldbereferredtoonlyafterattemptingtheproblems.
2 Someofthematerialisstudiedinsomecirclesandsummerschoolsbythosewho arejustgettingacquaintedwithmathematics(forexample, 6thgraders).However,the presentationisintendedforareaderwhoalreadyhasatleastaminimalmathematical background.Youngerstudentsneedadifferentapproach;see,forexample,[GIF].
Asarule,wepresentthe formulation ofabeautifulorimportantresult (intheformofproblems)beforeits proof.Insuchcases,onemayneed tosolvelaterproblemsinordertofullyworkouttheproof.Thisisalways explicitlymentionedinthetext.Consequently,ifyoufailtosolveaproblem, pleasereadon.Thisguidelineishelpfulbecauseitsimulatesthetypical researchsituation(see[ZSS,Ch.28]).
Thisbook“isanattempttodemonstratelearningas dialogue basedon solvinganddiscussingproblems”(see[KK15]).
Partingwords ByA.Ya.Kanel-Belov Tosolvedifficultolympiadproblems,attheveryleastonemusthavearobust knowledgeofalgebra(particularlyalgebraictransformations)andgeometry. Mostolympiadproblems(exceptfortheeasiestones)require“mixed”approaches;rarelyisaproblemresolvedbyapplyingamethodorideainits pureform.Approachingsuchmixedproblemsinvolvescombiningseveral “crux”problems,eachofwhichmayinvolvesingleideasina“pure”form.
Learningtomanipulatealgebraicexpressionsisessential.Thelackof thisskillamongolympiansoftenleadstoridiculousandannoyingmistakes.
Olympiadsandmathematics Tohimathinkingman’sjobwasnottodenyonereality attheexpenseoftheother,buttoincludeandtoconnect.
U.K.LeGuin. TheDispossessed.
Herearethreecommonmisconceptions:thebestwaytopreparefora matholympiadisbysolvinglastyear’sproblems;thebestwaytolearn “serious”mathematicsisbyreadinguniversitytextbooks;thebestwayto masteranyotherskilliswithnomathatall.Afurthermisconception isthatitdifficulttoachieveanytwoofthesethreegoalssimultaneously, becausetheyaresodivergent.Theauthorssharethebeliefthatthesethree approachesmissthepointandleadtoharmfulsideeffects:studentseither becometookeenonemulation,ortheystudythe language ofmathematics ratherthanits substance,ortheyunderestimatethevalueofrobustmath knowledgeinotherdisciplines.
Webelievethatthesethreegoalsarenotasdivergentastheymightseem. Thefoundationofmathematicaleducationshouldbethe solutionanddiscussionofproblemsinterestingtothestudent,duringwhichastudentlearns importantmathematicalfactsandconcepts. Thissimultaneouslyprepares thestudentformatholympiadsandthe“serious”studyofmathematics,and isgoodforhisorhergeneraldevelopment.Moreover,itismoreeffectivefor achievingsuccessinanyoneofthethreegoalsabove.
Researchproblemsforhighschoolstudents Manytalentedhighschooloruniversitystudentsareinterestedinsolving researchproblems.Suchproblemsareusuallyformulatedascomplexquestionsbrokenintoincrementalsteps;see,e.g.,[LKTG].Thefinalresultmay evenbeunknowninitially,appearingnaturallyonlyinthecourseofthinking abouttheproblem.Workingonsuchquestionsisusefulinitselfandisa goodapproximationtoscientificresearch.Thereforeitisusefulifateacher orabookcansupportanddevelopthisinterest.
Foradescriptionofsuccessfulexamplesofthisactivity,see,forexample,projectsintheMoscowMathematicalConferenceofHighSchoolStudents[M].Whilemostoftheseprojectsarenotcompletelyoriginal,sometimestheycanleadtonewresults.
Howthisbookisorganized Youshouldnotreadeachpageinthisbook,oneaftertheother.Youcan chooseasequenceofstudythatisconvenientforyou(oromitsometopics altogether).Anysection(orsubsection)ofthebookcanbeusedforamath circlesession.
Thebookisdividedintochaptersandsections(somesectionsaredivided intosubsections),withaplanoforganizationoutlinedatthestartofeach section.Ifthematerialofanothersectionisneededinaproblem,youcan eitherignoreitorlookupthereference.Thisallowsgreaterfreedomwhen studyingthebookbutatthesametimeitmayrequirecarefulattentiveness.
Thetopicsofeachchapterarearrangedapproximatelyinorderofincreasingcomplexity.Thenumbersinparenthesesafteratopicnameindicate its“relativelevel”:1isthesimplest,and4isthemostdifficult.Thefirst topics(notmarkedwithanasterisk)arebasic;unlessindicatedotherwise, youshouldbeginyourstudywiththem.Theremainingones(markedwith anasterisk)canbereturnedtolater;unlessotherwisestated,theyareindependentofeachother.Forproblemsmarkedwiththedegreesign(◦ ),it sufficestogivejustananswer,notnecessarilyaproof.Asyouread,tryto return tooldmaterial,butatanewlevel.Thusyoushouldendupstudying differentlevelsofatopic notsequentially butaspartofamixtureoftopics.
Thenotationusedthroughoutthebookisgivenonp.xvii.Notationand conventionsparticulartoaspecificsectionareintroducedatthebeginning ofthatsection.Thebookconcludeswithasubjectindex.Thenumbersin boldarethepagesonwhich formaldefinitions ofconceptsaregiven.
Resourcesandliterature Besidessourcesforspecializedmaterial,wehavealsotriedtoincludethevery bestpopularwritingonthetopicsstudied.Wehopethatthebibliography,at leastasafirstapproximation,canguidereadersthroughtheseaofpopular
NUMBERINGANDNOTATIONxvii
scientificliteratureinmathematics.However,thegreatsizeofthisgenre guaranteesthatmanynotableworkshadtobeomitted.Pleasenotethat sourcesinthebibliographyarenotnecessaryforsolvingtheproblemsinthis book,unlessexplicitlystatedotherwise.
Manyoftheproblemsarenotoriginal,butthesource(evenifitisknown) isusuallynotspecified.Whenareferenceisprovided,itcomesafterthe statementoftheproblem,sothatreaderscancomparetheirsolutionswith theonegiventhere.Whenweknowthatmanyproblemsinasectioncome fromonesource,wementionthis.
Wedonotprovidelinkstoonlineversionsofarticlesinthepopular magazines Quantum and MathematicalEnlightenment ;theycanbefoundat thewebsiteshttp://kvant.ras.ru,http://kvant.mccme.ru,andhttp://www. mccme.ru/free-books/matpros.html.
Acknowledgments Wearegratefulfortheseriousworkofthetranslatorsofthebook,PaulZeitz andSergeyShubin.Wethankthereviewersforhelpfulcomments,specifically,E.A.Avksentiev(Ch.3,Ch.8),E.V.Bakaev(Ch.6),V.N.Dubrovsky (Ch.3),A.A.Polyansky(Ch.4,Ch.7),G.I.Sharygin(Ch.1,Ch.2),and D.E.Shnol(Ch.5),aswellasanonymousreviewersofselectedmaterials. WethankA.I.Kanel-BelovandA.V.Shapovalov(http://www.ashap.info), theauthorsofagreatbodyofwork,whoalsocontributedanumberofuseful ideasandcomments.WealsothankD.A.Permyakov,theeditorofthebook [ZPSSS].Wethankourstudentsforaskingtrickyquestionsandpointingout errors.ThanksalsotoE.S.GorskyandP.V.Shirokovforpreparingmanyof thedrawings.Furtheracknowledgmentsforspecificitemsaregivendirectly inthetext.Weapologizeforanymistakes,andwouldbegratefultoreaders forpointingthemout.
M.B.SkopenkovwaspartiallysupportedbygrantsfromtheSimons FoundationandtheDynastyFoundation.
Contactinformation M.B.Skopenkov:NationalResearchUniversityHigherSchoolofEconomics (FacultyofMathematics)andtheInstituteforInformationTransmission ProblemsoftheRussianAcademyofSciences.
A.A.Zaslavsky:CentralEconomicMathematicalInstitute,MoscowPower EnergeticInstitute,andMoscowSchool1543.
Numberingandnotation Thetopicsineachchapterarearrangedapproximatelyinorderofincreasing complexityofthematerial.Thenumbersinparenthesesafterthetopicname indicateits“relativelevel”:1isthesimplest,and4isthemostdifficult.
Thefirsttopics(notmarkedwithanasterisk)arebasic;unlessindicated otherwise,youcanbegintostudythechapterwiththem.Therestofthe topics(markedwithanasterisk)canbereturnedtolater;unlessotherwise stated,theyareindependentofeachother.
Eachparagraphstartingwithanumber(suchas“1.1”)isaproblem. Ifthestatementoftheproblemisintheformofanassertion,thenthe problemistoprovethisassertion.Moreopen-endedquestionsarecalled challenges ;hereonemustcomeupwithaprecisestatementandaproof;cf., forexample,[VIN].
Themostdifficultproblemsaremarkedwithasterisks(*).Ifthestatementoftheproblemasksyouto“find”something,thenyouneedtogive a“closedform”answer(asopposedto,say,anunevaluatedsumofmany terms).Again,ifyouareunabletosolveaproblem,readon;laterproblems mayturnouttobehints.
Notation • x =[x] —(lower)integerpartofthenumber x (“floor”);thatis,the largestintegernotexceeding x.
• x —upperintegerpartofthenumber x (“ceiling”);thatis,thesmallest integernotlessthan x.
•{x} —fractionalpartofthenumber x;equalto x − x .
• d | n or n . . d d divides n;thatis; d =0 andthereexistsaninteger k suchthat n = kd (thenumber d iscalleda divisor ofthenumber n).
• R, Q,and Z —thesetsofallreal,rational,andintegernumbers,respectively.
• Z2 —theset {0, 1} ofremaindersupondivisionby2withtheoperations ofadditionandmultiplicationmodulo2.
• Zm —theset {0, 1,...,m 1} ofremaindersupondivisionby m withthe operationsofadditionandmultiplicationmodulo m.(Specialistsinalgebra oftenwritethissetas Z/mZ anduse Zm forthesetof m-adicintegers for theprime m.)
• n k —thenumberof k -elementsubsetsofan n-elementset(alsodenoted by C k n ).
•|X | —numberofelementsinset X
• A B = {x | x ∈ A and x/ ∈ B } —thedifferenceofthesets A and B
• A ⊂ B —meanstheset A iscontainedintheset B .Insomebooksthis isdenotedby A ⊆ B ,and A ⊂ B means“theset A isintheset B andisnot equalto B .”
• Weabbreviatethephrase“Define x by a”to x := a
References [GDI]A.A.Glibichuk,A.B.Dajnyak,D.G.Ilyinsky,A.B.Kupavsky,A.M.Rajgorodsky,A.B.Skopenkov,andA.A.Chernov, Elementsofdiscretemathematicsin problems,MCCME,Moscow2016(Russian). Abridgedversionavailableathttp://www.mccme.ru/circles/oim/discrbook.pdf.
[GIF]S.A.Genkin,I.V.Itenberg,andD.V.Fomin, Leningradmathematicalcircles, Kirov,1994(Russian).
[GKP]RonaldL.Graham,DonaldE.Knuth,andOrenPatashnik, Concretemathematics: Afoundationforcomputerscience,2nded.,Addison-WesleyPublishing Company,Reading,MA,1994.MR1397498
[VIN]O.Ya.Viro,O.A.Ivanov,N.Yu.Netsvetaev,andV.M.Kharlamov, Elementarytopology: problemtextbook,AmericanMathematicalSociety,Providence,RI, 2008,DOI10.1090/mbk/054.MR2444949
[ZSS]A.Zaslavsky,A.SkopenkovandM.Skopenkov(eds.), Elementsofmathematics inproblems:througholympiadsandcirclestotheprofession,MCCME,Moscow, 2017(Russian).
[ZPSSS]D.Permyakov,A.Shapovalov,A.Skopenkov,M.Skopenkov,andA.Zaslavskiy (eds.), Mathematicsinproblems.MaterialsofschoolsforMoscowteamtoRussianmathematicalolympiad,MCCME,Moscow,2009,(Russian).Shortversion availableathttp://www.mccme.ru/circles/oim/materials/sturm.pdf.
[Ar04]VladimirArnold, Problemsofchildren5to15yearsold,Eur.Math.Soc.Newsl. 98 (2015),14–20.Excerptfrom Lecturesandproblems:agifttoyoungmathematicians,AmericanMathematicalSociety,Providence,RI,2015[MR3409220]. MR3445185
[BSh]A.D.BlinkovandA.V.Shapovalov(eds.),Bookseries“Schoolmathcircles”.
[IBL]http://en.wikipedia.org/wiki/Inquiry-based_learning
[KK08]A.Ya.Kanel-BelovandA.K.Kovalji, Howtosolvenon-standardproblems, MCCME,Moscow2008(Russian).
[KK15]A.K.KovaljiandA.Ya.Kanel-Belov, MathematicsClasses—LeafletsandDialogue,MathematicalEducation 3 (2015),no.19,206–233(Russian).
[LKTG]SummerConferencesoftheTournamentofTowns,http://www.turgor.ru/en/ lktg/index.php.
[M]MoscowMathematicalConferenceofHighSchoolStudents,http://www.mccme. ru/mmks/index.htm.
[Mey]D.Meyer,StarterPack,http://blog.mrmeyer.com/starter-pack.
[PoSe]G.PólyaandG.Szegő, Problemsandtheoremsinanalysis.Vol.I:Series,integralcalculus,theoryoffunctions,Springer-Verlag,NewYork-Berlin,1972.TranslatedfromtheGermanbyD.Aeppli;DieGrundlehrendermathematischenWissenschaften,Band193.MR0344042
[Pr07-1]ViktorVasil evichPrasolov, Zadachipoplanimetrii.Chast I(Russian),BibliotekaMatematicheskogoKruzhka[LibraryfortheMathematicalCircle],vol.15, Nauka,Moscow,1986.MR966667
[Pl]Plato, Phaedo,Kindleedition,AmazonDigitalServices,2012.
[Ro04]V.A.Rokhlin, Lectureonteachingmathematicstononmathematicians,MathematicalEducation 3 (2004),no.8,21–36(Russian).
[RMP]RossMathematicsProgram,http://u.osu.edu/rossmath.
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xxINTRODUCTION [Sk09]A.B.Skopenkov, Fundamentalsofdifferentialgeometryininterestingproblems, MCCME,Moscow2009(Russian). arXiv:0801.1568
[Su]D.T.Suzuki, AnintroductiontoZenBuddhism,Kindleedition,AmazonDigital Services,2018.
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Chapter1 Triangle ThenotationforthischapterissummarizedinFig.1.
Figure1
• Let ABC beagiventriangle.Conventionally, Ai ,Bi ,and Ci ,i = 1, 2,..., denotepointsonthesides BC , CA,and AB respectively(or onextensionsofthesesides,ifrelevant).
21.TRIANGLE
• Let ω denotetheinscribedcircle(alsocalledthe“incircle”),withcenter I andradius r (alsocalledthe“inradius”).
• Let Ω denotethecircumscribedcircle(alsocalledthe“circumcircle”), withcenter O andradius R (alsocalledthe“circumradius”).
• Definethe medians tobethesegmentsjoiningtheverticeswiththe midpointsoftheoppositesidesofthetriangle.Let G denotethepoint ofintersectionofthemedians(alsoknownasthe centerofgravity,or centroid ).
• Let H denotetheintersectionpointoftheheightsofthetriangle(also knownasthe orthocenter ).
• Extendtheanglebisectors AI , BI , CI sothattheyintersectwith Ω at A , B , C respectively.Thus, A , B , C arethemidpointsofthearcs AB , BC , CA respectively.
• The orthotriangle of ABC isthetrianglewhoseverticesarethefeetof theperpendicularsfrom A, B ,and C tolines BC , AC ,and AB (i.e., theheights).
• The medialtriangle of ABC isthetrianglewhoseverticesarethemidpointsofthesides.Thesidesofthemedialtrianglearethe midlines of thetriangle ABC .
• Bythe circle ABC wemeanthecircumscribedcircleofthetriangle ABC .
• Bya circularsegment wemeanafigureboundedbyacircluararcand achord.
• An externalbisector isthebisectorofoneoftheexternalanglesofa triangle.
• Denoteby h(A,BC ) theperpendiculardroppedfromthepoint A to theline BC .
• Threeormorelinesare concurrent iftheyhaveauniquecommonpoint. Threeormorepointsare collinear iftheybelongtooneline.
Somenotabletopicsinthegeometryofatrianglenotcoveredby thecontentofthischaptercanbefoundin[EM90, Sh89, Ku92].
1.Carnot’sprinciple(1) ByV.Yu.ProtasovandA.A.Gavrilyuk
1.1.1.Carnot’sTheorem. Let A1 , B1 , C1 bepointslyingonthesidesor extensionsofthesidesoftriangle ABC .Drawlinesthroughthesepoints perpendiculartothesidestheyareon.Provethatthelinesintersectata singlepointifandonlyif
1.1.2. FormulateandproveageneralizedCarnot’stheoremforarbitrary pointsintheplane A1 , B1 , C1 ,notnecessarilylyingonlinescontainingthe sidesofatriangle ABC
1.1.3.◦ Forwhichofthefollowingcasesisitpossiblethattheperpendiculars drawntothesides(orextensionsofsides)ofthetrianglethroughthespecified pointsmaynotbeconcurrent?
1) A1 , B1 , C1 arethepointsoftangencyofthesidesoftriangle ABC to theinscribedcircle.
2) A2 , B2 , C2 arethepointsoftangencyofthesidesoftriangle ABC to thecorrespondingexcircles.
3) A3 , B3 , C3 aretheintersectionsoftheanglebisectorswiththecorrespondingsidesofthetriangle.
1.1.4. Considerthreeintersectingcirclesintheplane.Provethattheirthree pairwisecommonchordsareconcurrent.
Note: Thisstatementisusuallyprovenusingthenotionofthepower ofapoint(seeProblem2.4.1).However,itiseasytoderiveitfromthe generalizedCarnot’stheorem.
1.1.5. UseCarnot’sprincipletogiveyetanotherproofthattheheightsofa triangleareconcurrent.
1.1.6. Oneachsideofatriangle,considerthepointofintersectionofthe anglebisectoroftheangleoppositethisside.Thendrawtheperpendicular tothissidethroughthisintersectionpoint.Describealltrianglesforwhich thesethreeperpendicularsareconcurrent.
1.1.7. Onthesidesoftriangle ABC ,constructtherectangles ABB1 A1 , BCC2 B2 ,and CAA2 C1 .Provethattheperpendicularbisectorstosegments A1 A2 , B1 B2 ,and C1 C2 areeitherconcurrentorparallel.
1.1.8. Let A1 , B1 , C1 bethemidpointsofsides BC , AC , AB oftriangle ABC ,respectively.Let B2 bethebaseoftheperpendiculardroppedfrom B to A1 C1 .Define A2 and C2 similarly.Provethat h(C1 ,A2 B2 ), h(B1 ,A2 C2 ), and h(A1 ,B2 C2 ) areconcurrent.
1.1.9. Provethat h(A,B1 C1 ), h(B,A1 C1 ),and h(C,A1 B1 ) areconcurrent ifandonlyif h(A1 ,BC ), h(B1 ,AC ),and h(C1 ,AB ) areconcurrent.
1.1.10. Givenequilateraltriangle ABC andapoint D thatdoesnotlieon lines AB , AC ,or BC ,let A1 betheincenteroftriangle BCD ,with B1 and C1 definedsimilarly.Provethat h(A,B1 C1 ), h(B,A1 C1 ),and h(C,A1 B1 ) areconcurrent.
Suggestions,solutions,andanswers
1.1.1. Lettheperpendicularsdrawnthroughpoints A1 and B1 intersectat point M .ApplyingthePythagoreanTheoremtotherighttriangles AMB1 and CMB1 yields |B1 A|2 −|B1 C |2 = |MA|2 −|MC |2 .(Thistechnique,
41.TRIANGLE wherewereplacethedifferenceofthesquaresofthehypotenusesbythe differenceofthesquaresoftheirlegs,iscalled Carnot’sprinciple.)
Nowlettheperpendicularstothesidesofthetriangledrawnthrough points A1 , B1 , C1 intersectatpoint M .ApplyingCarnot’sprinciple,we obtaintherequiredequality.
Conversely,letpoints A1 , B1 , C1 besuchthat
Let M denotethepointofintersectionoftheperpendicularsdrawnthrough A1 and B1 tothecorrespondingsides,andconsidertheperpendicular MC droppedfrom M to AB .Asabove,wehave
andthus C coincideswith C1 ,whichiswhatshouldbeproved.
1.1.5. Applythestatementofthepreviousproblemtothreecircleswhose diametersarethesidesofthetriangle,ordirectlyapplyCarnot’sTheorem tothefeetoftheheights.
1.1.7. Tobeginwith,notethattheperpendicularsdroppedfrom A, B , C tolines A1 A2 , B1 B2 , C1 C2 ,respectively,areconcurrent.Indeed,for example,aperpendiculardroppedfrom C to C1 C2 dividestheangle C into twoanglesequaltotheanglesoftriangle CC1 C2 ,whosesinesareintheratio |CC1 |/|CC2 |.UsingsimilarrelationsforthetwootherverticesandCeva’s Theorem,weobtaintherequiredstatement.
Nowsupposethatlines A1 A2 , B1 B2 ,and C1 C2 formatriangle A B C , andlinesparalleltothempassingthrough A, B ,and C formthetriangle A B C (thecasewheretwoofthelines A1 A2 , B1 B2 ,and C1 C2 areparallel iseasier).Then,forexample,triangles A1 B1 C and ABC arecongruentby SAA,andthusif B0 isthemidpointof B1 B2 ,then
ApplyingCarnot’sprincipletothesimilartriangles A B C and A B C yieldsthestatementoftheproblem.
2.Thecenteroftheinscribedcircle(2) ByV.Yu.Protasov
Theproblemsofthissectionaresimilarincontenttoproblemsfromsections 3,5,and7ofthischapter(“TheEulerline”,“Theorthocenter,orthotriangle, andnine-pointcircle”,and“Bisectors,heights,andcircumcircles”,respectively).
1.2.1.◦ Choosethecorrectanswer:Angle AIB isequalto 1) π ∠C ;2) (π + ∠C )/2;3) 2∠C
1.2.2. Let ABCD beacyclicquadrilateral.Provethattheincentersof triangles ABC , BCD , CDA, DAB aretheverticesofarectangle.
1.2.3. Let ABCD beacyclicquadrilateral.Provethatthesumoftheradii oftheinscribedcirclesoftriangles ABC and CDA isequaltothesumof theradiioftheinscribedcirclesoftriangles BCD and DAB .
1.2.4. Throughpoint M insideagiventriangle,threecongruentcirclesare drawn,eachofwhichistangenttotwosidesofthetriangle.
(a)Provethat M liesonthelineconnectingthecentersoftheinscribed andcircumscribedcirclesofthetriangle.
(b)Describeamethodforconstructingsuchapoint M ,giventhetriangle.
(c)Let x betheradiusofthecongruentcircles.Provethat 2r 3 ≤ x ≤ R 3 . Isittruethatifoneoftheseinequalitiesisanequality,thenthetriangleis equilateral?
(d)Prove Euler’sinequality : R ≥ 2r .Forwhichtrianglesisitanequality?
1.2.5. Eachofthreecongruentcirclesistangenttotwosidesofatriangle, andafourthcircle,withthesameradius,istangenttothesethreecircles.
(a)Provethatthecenterofthefourthcircleliesonalineconnectingthe centersoftheinscribedandcircumscribedcirclesofthetriangle.
(b)Describehowtoconstructsuchcirclesforthegiventriangle.
(c)Expresstheradiusofthecongruentcirclesintermsof r and R.
1.2.6. Letatriangle ABC bescalene,i.e.,allthreesideshavedifferent lengths.Denoteby A0 , B0 , C0 themidpointsofitssides.Theinscribed circleistangenttoside BC atpoint A1 ,andpoint A2 issymmetricto A1 withrespecttothebisectorofangle A.Definepoints B2 and C2 similarly. Provethat A0 A2 , B0 B2 ,and C0 C2 areconcurrent.
Suggestions,solutions,andanswers
1.2.2. Bythetridentlemma(Problem1.7.3)wehave |C A| = |C I | = |C B | Nowlet Ia , Ib , Ic , Id bethecentersoftheinscribedcirclesoftriangles BCD , CDA, DAB , ABC respectively.Thenpoints A, B , Ic , Id lieon onecircle;therefore ∠BId Ic = π ∠BAIc = π ∠BAD/2.Similarly, ∠BId Ia = π ∠BCD/2.So ∠Ia Id Ic =(∠BAD + ∠BCD )/2= π/2
1.2.4. (a) Suggestion.Considerahomothety(seethedefinitioninsection4 ofChapter3)withacenteratpoint I Pathtosolution (N.Medved) Wehaveanarbitrarytriangle ABC .Let Oa , Ob ,and Oc bethecentersofthecongruentcircles a, b,and c,and
61.TRIANGLE
let I and O bethecentersoftheinscribedandcircumscribedcirclesof triangle ABC ,respectively.Drawtheradiifromthecentersof a, b, c to thepointsoftangencywithtriangle ABC .Theseradiiareequal(thecircles arecongruent),andwhendroppedtothesamesidewillbeparallel(the sidesofthetrianglearetangenttocircles).Sobyconnectingpoints Oa , Ob , Oc anddrawingfromthemradiitopointsoftangencywiththesides ofthetriangle,wegetthreerectangles;i.e.,triangle ABC willbesimilar totriangle Oa Ob Oc ,withthesidesoftriangle ABC beingparalleltothe correspondingsidesoftriangle Oa Ob Oc .Furthermore,notethatlines AOa , BOb ,and COc intersectat I ,sincetheycoincidewiththeanglebisectorsof triangle ABC .Consequently,thereexistsahomothetywithcenter I that transformstriangle ABC intotriangle Oa Ob Oc .Thishomothetymoves O to thecenterofthecircumcircleoftriangle Oa Ob Oc ,thatis,to M (since Oa M , Ob M ,and Oc M aretheradiiofequalcircles a, b,and c).Therefore,points I , O ,and M arecollinear.
(b)(N.Medved)Fromthesolutionof(a),weknowthattherequired point M isthecenterofthecircumscribedcircleoftriangle Oa Ob Oc ;i.e., tosolvetheproblemitsufficestoconstructtriangle Oa Ob Oc ,andthenitis easytofindthecenterofitscircumscribedcircle,whichispoint M .
Startbynotingthatwecanconstructtheradiioftheinscribedandthe circumscribedcirclesoftriangle ABC .Simplyconstructtheanglebisectors andconnecttheirpointofintersectionwithatangentpointinthetriangle;thissegmentwillbetheradiusoftheinscribedcircleoftriangle ABC . Forthecircumcircle,constructtheperpendicularbisectorsofthesidesand connecttheirintersectionpointwithoneoftheverticesofthetriangle;this segmentwillbetheradiusofcircumcircle ABC .
From(a),weseethat x R = r x r ,where x istheradiusofthecircumcircle oftriangle Oa Ob Oc .Thisimplies x = Rr R+r ,whence x r = R R+r .Toconstruct asegmentoflength x,drawanarbitraryangle A1 ,ononesideofwhichwe markpoint B1 suchthat |A1 B1 | = R;ontheothersidewemarkpoints C1 and D sothat |A1 C1 | = r and |C1 D | = R.Next,connectpoints B1 and D togetsegment B1 D .Drawaparallelsegmentfrom C1 thatintersectsthe sideofangle A1 atpoint F .Thelengthofsegment A1 F willequal x
Drawthreelinesparalleltothesidesofthetriangle,atadistanceof x fromeachside.Thepointsoftheirintersectionwillbetheverticesofthe desiredtriangle Oa Ob Oc .Finally,constructtheperpendicularbisectorsof thesidesofthistriangle;theirpointofintersectionwillbethedesiredpoint M .
(c)Considerhomotheticimagesofthegiventrianglewithratio 2 3 ,with centersattheverticesofthegiventriangle.Thesethreetriangleshavea singlecommonpoint.Fromthis,itfollowsthat x cannotbelessthan 2r 3 . Further,fromthesimilarityoftheoriginaltriangleandthetrianglewith verticesatthecentersofthegivencircles,itfollowsthat x R = r x r =1 x r ≤
1 2 3 = 1 3 .
3.TheEulerline ByV.Yu.Protasov
Theproblemsofthissectioncoverthesametopicsasproblemsinsections 2,5,and7ofthischapter.
1.3.1. Inanytriangle,points O,G,and H lieononeline(calledthe Euler line ),and |GH | =2 ·|GO |
1.3.2.◦ TheEulerlineofanon-isoscelestrianglepassesthroughoneofits vertices.Whatistheangleatthisvertex?
1) 90◦ 2) 120◦ 3) 60◦ 4)Therearenosuchtriangles.
1.3.3. ProvethattheEulerlineisparalleltoside AB ifandonlyif tan A tan B =3
1.3.4. TheEulerlineofatriangleisparalleltooneofitsanglebisectors. Provethateitherthetriangleisisosceles,oroneofitsanglesis 120◦
1.3.5. Let ∠A =120◦ .Provethat |OH | = |AB | + |AC |
1.3.6. Foreachvertexofatriangle,constructthecirclethatgoesthrough thisvertexandthefootoftheheightfromthisvertexandwhichistangent totheradiusofthecircumcircledrawntothisvertex.Provethatthesethree circlesintersectattwopointswhichlieontheEulerlineofthetriangle.
1.3.7. Assumethatallanglesoftriangle ABC arelessthan 120◦ .Define the Torricellipoint of ABC tobethepoint T satisfying ∠ATB = ∠BTC = ∠CTA =120◦
(a)ProvethattheEulerlineoftriangle ATB isparalleltoline CT Suggestion.UseProblem1.3.3.
(b)ProvethattheEulerlinesoftriangles ATB , BTC ,and CTA are concurrent.
1.3.8. Attheverticesofanacutetriangle,drawtangentstoitscircumcircle. Provethatthecenterofthecircumscribedcircleofthetriangleformedby thesethreetangentsliesontheEulerlineoftheoriginaltriangle.
Suggestions,solutions,andanswers 1.3.3. Angle C mustbeacute,sinceotherwisepoints O and H lieonoppositesidesof AB .Sincethedistancefrom O to AB is R cos C andthe heightdrawnfromvertex C is |AC | sin A =2R sin A sin B ,weseethat theparallelismoftheEulerlineandline AB isequivalenttotheequality 3cos C =2sin A sin B. Theresultfollowsfrom cos C = cos(A + B )= sin A sin B cos A cos B
1.3.6. Itfollowsfromthestatementoftheproblemthatthepowerofthe point O relativetothesecirclesisequalto R2 .Inaddition,if AA and BB areheightsofthetriangle,thenthequadrilateral ABA B iscyclic,sothat |HA|·|HA | = |HB |·|HB |.Therefore,thepowersofthepoint H relative toallthreecirclesarealsoequal,i.e.,line OH istheircommonradicalaxis.
4.Carnot’sformula(2∗ ) ByA.D.Blinkov
Carnot’sformula (namedaftertheFrenchmathematician,physicist,and politicianLazareCarnot,1753–1823)statesthatinanacutetriangle,the sumofthedistancesfromthecenterofthecircumscribedcircletothesides ofthetriangleisequaltothesumoftheradiiofthecircumscribedand inscribedcircles: |OM1 | + |OM2 | + |OM3 | = R + r ,where M1 , M2 , M3 arethemidpointsof BC , CA, AB ,respectively.AproofusingPtolemy’s Theoremisgiveninsection6ofChapter2.Herewelookatitsapplications andanalternativeproof;theseinvestigationswillyieldotherimportantfacts.
1.4.1. Letthebisectorofangle A intersectthecircumcircleoftriangle ABC atpoint W ,andlet D bediametricallyoppositetopoint W .Provethat
(a) |M1 W | =(ra r )/2; (b) |M1 D | =(rb + rc )/2,where r istheradiusofinscribedcircleand ra , rb , rc aretheradiioftheexcircles.
1.4.2. ProveCarnot’sformula.
ForthefollowingproblemsthatexploreapplicationsofCarnot’sformula, assumeunlessotherwisestatedthatthetrianglesareacute.
1.4.3. Provethatthesumofthedistancesfromtheverticesofatriangle totheorthocenterisequaltothesumofthediametersofitsinscribedand circumscribedcircles.
1.4.4. Provethatintriangle ABC thefollowinginequalitieshold:
(a) |AH | + |BH | + |CH |≤ 3R; (b) 3|OH |≥ R 2r
1.4.5. (a)Provethat ma + mb + mc ≤ 9 2 R,where ma , mb ,and mc arethe lengthsofthemediansofthetriangle.
(b)Letthebisectorsofangles A, B ,and C intriangle ABC intersect thecircumcircleatpoints W1 , W2 ,and W3 respectively.Provethat |AW1 | + |BW2 | + |CW3 |≤ 6 5R r
1.4.6. (a)Provethattheanglesofatriangle ABC satisfytheinequality 3r R ≤ cos A +cos B +cos C ≤ 3 2
(b)Let AH1 , BH2 ,and CH3 betheheightsoftriangle ABC .Expressthe sumofthediametersofthecirclescircumscribedaroundtriangles AH2 H3 , BH1 H3 ,and CH1 H2 intermsof R and r
1.4.7. Atriangleisinscribedinacircleofradius R.Ineachsegmentofthe circleboundedbyasideofthetriangleandthelesserofthecirculararcs, inscribeacircleofmaximumradius.Findthesumofthediametersofthese threecirclesandtheradiusoftheincircleofthetriangle.
1.4.8. (a)Provethatintriangle ABC ,thefollowingequalityholds: a(|OM2 | + |OM3 |)+ b(|OM1 | + |OM3 |)+ c(|OM1 | + |OM2 |)=2pR where 2p istheperimeterof ABC
(b) Erdős’sinequality. Let ha bethegreatestheightoftriangle ABC Provethat ha ≥ R + r
1.4.9. (a)DeriveanaloguesofCarnot’sformulaforrightandobtusetriangles.
(b)Quadrilateral ABCD iscyclic.Let r1 and r2 betheradiiofcircles inscribedintriangles ABC and ADC ,andlet r3 and r4 betheradiiofthe circlesinscribedintriangles ABD and CBD .Provethat r1 + r2 = r3 + r4
1.4.10. Let d, d1 , d2 ,and d3 bethedistancesfromthecenter O ofthe circumcircleofatriangletothecentersofitsinscribedandexscribedcircles.
Provethat
1.4.11. (a)Provethatifapointbelongstothesegmentconnectingthebases oftwoanglebisectorsofatriangle,thenthesumofthedistancesfromthis pointtotwosidesofthetriangleisequaltothedistancefromittothethird side.
(b)Supposethecircumcenterofatriangleliesonthesegmentconnecting thebasesoftwoanglebisectors.Provethatthedistancefromtheorthocenter ofthistriangletooneofitsverticesisequalto R + r
Suggestions,solutions,andanswers 1.4.1. (a)Letpoints I and Ia ,respectively,bethecentersoftheinscribed circleandtheexcircletangenttoside BC .Let K and P bethepointsof tangencyofthesecircleswith BC ,andlet L betheintersectionpointof Ia P withthelinethatpassesthrough I whichisparallelto BC .Let Q bethe midpointof IL.Since W isthemidpointofsegment IIa (aconsequenceof thetridentlemma;seeProblem1.7.3insection7)and WM1 IK LIa ,it followsthatsegment WQ isamidlineoftriangle ILIa ,and M1 lieson WQ Therefore, |WQ| = |Ia L|/2=(ra + r )/2,andthen |M1 W | = |WQ|− r = (ra r )/2
