Table Of ContentEllina Grigorieva
Methods of
Solving Complex
Geometry
Problems
Ellina Grigorieva
Methods of Solving Complex
Geometry Problems
EllinaGrigorieva
MathematicsandComputerScience
TexasWoman’sUniversity
Denton,TX,USA
ISBN978-3-319-00704-5 ISBN978-3-319-00705-2(eBook)
DOI10.1007/978-3-319-00705-2
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013943010
MathematicsSubjectClassification(2010):97G40,97D50,68T20,51N20
©SpringerInternationalPublishingSwitzerland2013
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To my beautiful daughter,
Sasha
And to my wonderful parents,
Natali and Valery Grigoriev
Your encouragement made this book possible
And to my university mentor and scientific
advisor academician,
Stepanov Nikolay Fedorovich
Without your help and brilliant mind my
career as a scientist would not have been
successful!
Preface
Many of us fear mathematics. Sometimes we live with this fear all our lives.
We assume that math is not for us and believe that our next-door neighbor is a
genius just because he or she is a math major. When I was young, I trained
professionally as a musician. One day in the sixth grade after missing 2 weeks of
schoolandcomingbackfromanationalviolincompetition,Ididnotdowellona
planegeometrytestbecauseIcouldnotunderstandhowtodoproofs.Ireceiveda
B(cid:1)onthetest—agreatdisappointment.Iaskedtheteacherwhy.Mymathteacher
told me, in front of the class, that I had no talent for the subject and that I should
continuepracticingtheviolin.Now,beingateachermyself,Iunderstandthatthis
approach was not the best pedagogical “trick” that can be applied in a classroom.
However, it worked in my case and it motivated me. I went home and worked
until I could solve all the problems in the book. It is important to mention that
before I was publicly criticized by my math teacher I did not realize that my
understanding of the subject was not clear. I remember sitting in the back of the
classandwatchinghowotherstudentswenttotheboardtoperformproofs.Ihadno
ideahowtheydidit.
Iwasalwaysgoodinalgebrabutgeometrywasdifferent.Geometrywasnotjust
thecalculationofsomethingortheapplicationofaformula.Ingeometryclasswe
hadtodoproofsusingEuclideanpostulatesandaxioms.Ihadneverdonesuchthings
before.ItwasevenmoreshockingformesinceImissedthefirsttwoweeksofthe
subject.Ididnotknowhowtoevenstartaproof.IhatedgeometryasstronglyasI
hatedplayingviolininthefirstgradewhennothingwasgoodenoughformyteacher.
ThatteacherdidnotwanttoworkwithmeanymorebecauseshealsobelievedthatI
didnothavetalentfortheviolinandsentmehomecrying.Iwassevenyearsoldand
didnotliketheviolinuntilIgotanewviolinteacher,Mr.London.Hewasableto
givemehopeanddevelopedmyskillssoIbecamethebestinmyviolinclass.Idid
notlikegeometryinthesameway.IdidnotunderstanditandIdidnotlikeit.But,
when I made an effort and solved every problem in the book I started to love the
subject.Wealwayslikethingsthatwearegoodatorthethingsthatcomeeasy.We
love something if we have visible, continuous success. Such success comes only
withhardwork.IlaterwentontowinvariousMathOlympiadsandgraduatefrom
vii
viii Preface
Lomonosov Moscow State University (MGU) summa cum laude, defend a Ph.D.
inMathematicaland Physical Sciences, and publish over 60papers inthe field of
differentialequations,gametheory,economics,andoptimalcontroltheory.
AtMoscowState,attheendofeachsemesterthestudentshadtopassfouroral
examsgivenbyrenownedprofessors.Theprofessorscouldaskanytrickyquestions
onthetopicoftheexaminationticket.WhenIwaspreparingforsuchexamsandin
order to have an “A,”I did nottry to memorize allthe definitions and proofs,but
rather I tried to develop a “global” understanding of the subject. I thought of
possible questions that an examiner could ask me and tried to predict the type
ofaproblemthatIcouldbeaskedtosolve.Idevelopedmyownwayoflearningand
Iwanttoshareitwithyou.Ifyouarestrugglingwithmath,thisbookisforyou.
Forover30yearswheneverIspottedanespeciallyinterestingortrickyproblem,
I added it to my notebooks along with my original solutions. I’ve accumulated
thousands of these problems. I use them every day in my teaching and included
manyoftheminthisbook.Pleaselookatoneofmynotebookpagesfromtheninth
grade in the Appendix. You will see how carefully and neatly each problem was
presented and solved. Please notice the importance of making an accurate sketch.
Many of the problems were created when I was an assistant professor and also
worked as a tutor to prepare high school students for the MGU entrance exams.
Before accepting an academic position at a university, I worked as a teacher at
Ursuline Academy of Dallas and used my problem-solving techniques in my
students’ college preparation. I was pleased to receive appreciation letters from
MITandHarvardwheresomeofmystudentswereadmitted.
If students see an elegant solution but do not apply the approach to other
problems, they will not remember it just as nobody remembers phone numbers
thesedays.However,ifateacherusesandreusesthesameapproachthroughoutthe
entire curriculum, students will remember it and learn to value the beauty of the
method. Most math books start from theoretical facts, give one or two examples,
and then a set of problems. In this book almost every statement is followed by
problems. You are not just memorizing a theorem; you apply the knowledge
immediately.Uponseeingasimilarprobleminthehomeworksectionyouwillbe
abletorecognizeandsolveit.
Although each section of this book can be studied independently, this book is
constructed to reinforce patterns developed at stages throughout this book. This
helps you to see how math topics are connected. This book can be helpful for
self-education, for people who want to do well in math classes or to prepare
forcompetitions.Thisbookisalsomeantformathteachersandcollegeprofessors
whowouldliketouseitasanextraresourceintheirclassroom.
Preface ix
What Is This Book About?
Geometry is a very important field of mathematics with applications in science,
architecture,andart.Itisdifficulttofindafieldofhumanactivitywheregeometry
is not applied. People build houses and office buildings, roads, and airplanes.
Geometry is everywhere. Even if you are not going to become a scientist or
engineer and wish only to graduate from a university, without knowledge of
geometryyoumaynotpasstheSAT.
Geometryisprobablytheoldestpartofmathematics.Ancientpeoplewantedto
know how big their property was, its capacity for growing crops, and how much
fencewasneeded.Aroundthattimesimplegeometricfigureswereintroducedsuch
astriangles,squares,andcircles.ProblemssolvedbyancientGreeksandEgyptians
areveryunusualandchallengingandtheirdifferentversionsappearonsomeexams
andmathcontests.ReviewingoverlookedproblemsandtheirsolutionsthatIoffer
in this book can prepare you for solving Math Olympiad problems. Few teachers
cover such problems. Some teachers do not have time to explain a strategy for
solving these problems; others do not have the background. This book is not a
textbook.Thisbookdoesnotcovereverytopicingeometry,butitwillprovideyou
with a brief course in plane geometry and it will help you to develop problem-
solvingskills.Itwillhelpyoutoimproveyourmathematicalabilities.
This book is briefly divided into four chapters: Triangle, Quadrilaterals and
Polygons,Circles,andProblemsonConstruction.Eachchapterhasitsownhome-
work. However, there are overlaps between chapters, because, for example, a
problem on a triangle and a circle inscribed into it cannot be solved without
knowledgeofthepropertiesofeach.Sometimesforthesamereason,inasolution
orproofwewilluseaproperty(lemmaortheorem)proveninalatersection.Soif
youknowthatpropertyyoucanfollowalongrightawayand,ifnot,thenyoumay
finditinthefollowingsectionsorinthesuggestedreferences.
It is obvious that two distinct points form a unique line. How do you find out
whetherornotthreegivenpointsareonthesameline(collinear)?Itiswellknown
(but youwilllearn how toprove it)thatthree mediansandthreealtitudesconcur.
WhatifwetakesomepointsM,N,andKonthesidesofatriangleandconnectthem
withthecorrespondingoppositevertex?Underwhatconditionwillsuchsegments
(cevians) intersect at one point (concur)? The answer to these and many other
questionswillbefoundinthisbook.Thisbookcontainsimportantandsometimes
overlooked topics on triangles, quadrilaterals, and circles such as Menelaus-Ceva
theorems, Simson line, theorems of the three altitudes and medians, and Heron’s
formulaalongwiththeirproofs.YouwillbeabletodissectasegmentintheGolden
ratio, construct an angle of 36(cid:3), and visualize Fibonacci numbers. You will prove
the inequality between geometric, arithmetic, and harmonic means in a purely
geometricway:thewayitwasdone1000yearsagobyancientGreeks,andprove
whynoteveryanglecanbetrisectedusingmodernmethods.Youwilllearnwhich
quadrilateral with diagonals d and d has the maximum area and solve unusual
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