Euclidean Geometry in Mathematical Olympiads

328 Pages·2016·5.39 MB·English

by  Evan Chen

Euclidean Geometry in
Mathematical Olympiads
With 248 Illustrations
(cid:2)c 2016by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressControlNumber:2016933605
PrintISBN:978-0-88385-839-4
ElectronicISBN:978-1-61444-411-4
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10 9 8 7 6 5 4 3 2 1
Euclidean Geometry in
Mathematical Olympiads
With 248 Illustrations
Evan Chen
PublishedandDistributedby
TheMathematicalAssociationofAmerica
CouncilonPublicationsandCommunications
JenniferJ.Quinn,Chair
CommitteeonBooks
JenniferJ.Quinn,Chair
MAAProblemBooksEditorialBoard
GailSNelson,Editor
ClaudiAlsina
ScottAnnin
AdamH.Berliner
JenniferRocheBowen
DouglasB.Meade
JohnH.Rickert
ZsuzsannaSzaniszlo
EricR.Westlund
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Aha!Solutions,MartinErickson
The Alberta High School Math Competitions 1957–2006: A Canadian Problem Book,
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TheContestProblemBookVII:AmericanMathematicsCompetitions,1995–2000Contests,
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EuclideanGeometryinMathematicalOlympiads,byEvanChen
First Steps for Math Olympians: Using the American Mathematics Competitions, by J.
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A Friendly Mathematics Competition: 35 Years of Teamwork in Indiana, edited by Rick
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AGentleIntroductiontotheAmericanInvitationalMathematicsExam,byScottA.Annin
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TheInquisitiveProblemSolver,PaulVaderlind,RichardK.Guy,andLorenC.Larson
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Mathematical Olympiads 1998–1999: Problems and Solutions From Around the World,
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A Mathematical Orchard: Problems and Solutions, by Mark I. Krusemeyer, George T.
Gilbert,andLorenC.Larson
ProblemsfromMurrayKlamkin:TheCanadianCollection,editedbyAndyLiuandBruce
Shawyer
Trigonometry:ACleverStudyGuide,byJamesTanton
The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938–
1964,A.M.Gleason,R.E.Greenwood,L.M.Kelly
The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965–
1984,GeraldL.Alexanderson,LeonardF.Klosinski,andLorenC.Larson
TheWilliamLowellPutnamMathematicalCompetition1985–2000:Problems,Solutions,
andCommentary,KiranS.Kedlaya,BjornPoonen,RaviVakil
USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and
ZumingFeng
USA and International Mathematical Olympiads 2001, edited by Titu Andreescu and
ZumingFeng
USA and International Mathematical Olympiads 2002, edited by Titu Andreescu and
ZumingFeng
USA and International Mathematical Olympiads 2003, edited by Titu Andreescu and
ZumingFeng
USAandInternationalMathematicalOlympiads2004,editedbyTituAndreescu,Zuming
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Dedicatedtothe
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Contents
Preface xi
Preliminaries xiii
0.1 TheStructureofThisBook............................................xiii
0.2 CentersofaTriangle ................................................. xiv
0.3 OtherNotationsandConventions...................................... xv
I Fundamentals................................................ 1
1 AngleChasing 3
1.1 TrianglesandCircles................................................. 3
1.2 CyclicQuadrilaterals................................................. 6
1.3 TheOrthicTriangle................................................... 7
1.4 TheIncenter/ExcenterLemma......................................... 9
1.5 DirectedAngles...................................................... 11
1.6 TangentstoCirclesandPhantomPoints................................ 15
1.7 SolvingaProblemfromtheIMOShortlist.............................. 16
1.8 Problems............................................................ 18
2 Circles 23
2.1 OrientationsofSimilarTriangles....................................... 23
2.2 PowerofaPoint ..................................................... 24
2.3 TheRadicalAxisandRadicalCenter................................... 26
2.4 CoaxialCircles....................................................... 30
2.5 RevisitingTangents:TheIncenter...................................... 31
2.6 TheExcircles........................................................ 32
2.7 ExampleProblems................................................... 34
2.8 Problems............................................................ 39
3 LengthsandRatios 43
3.1 TheExtendedLawofSines........................................... 43
3.2 Ceva’sTheorem...................................................... 44
3.3 DirectedLengthsandMenelaus’sTheorem............................. 46
3.4 TheCentroidandtheMedialTriangle.................................. 48
vii
viii Contents
3.5 HomothetyandtheNine-PointCircle................................... 49
3.6 ExampleProblems................................................... 51
3.7 Problems............................................................ 56
4 AssortedConfigurations 59
4.1 SimsonLinesRevisited............................................... 59
4.2 IncirclesandExcircles................................................ 60
4.3 MidpointsofAltitudes................................................ 62
4.4 EvenMoreIncircleandIncenterConfigurations......................... 63
4.5 IsogonalandIsotomicConjugates...................................... 63
4.6 Symmedians......................................................... 64
4.7 CirclesInscribedinSegments......................................... 66
4.8 MixtilinearIncircles.................................................. 68
4.9 Problems............................................................ 70
II Analytic Techniques......................................... 73
5 ComputationalGeometry 75
5.1 CartesianCoordinates ................................................ 75
5.2 Areas............................................................... 77
5.3 Trigonometry........................................................ 79
5.4 Ptolemy’sTheorem................................................... 81
5.5 ExampleProblems................................................... 84
5.6 Problems............................................................ 91
6 ComplexNumbers 95
6.1 WhatisaComplexNumber?.......................................... 95
6.2 AddingandMultiplyingComplexNumbers............................. 96
6.3 CollinearityandPerpendicularity...................................... 99
6.4 TheUnitCircle......................................................100
6.5 UsefulFormulas.....................................................103
6.6 ComplexIncenterandCircumcenter ................................... 106
6.7 ExampleProblems...................................................108
6.8 When(Not)touseComplexNumbers..................................115
6.9 Problems............................................................115
7 BarycentricCoordinates 119
7.1 DefinitionsandFirstTheorems........................................119
7.2 CentersoftheTriangle................................................122
7.3 Collinearity,Concurrence,andPointsatInfinity.........................123
7.4 DisplacementVectors.................................................126
7.5 ADemonstrationfromtheIMOShortlist...............................129
7.6 Conway’sNotations..................................................132
7.7 DisplacementVectors,Continued......................................133
7.8 MoreExamples......................................................135
7.9 When(Not)toUseBarycentricCoordinates.............................142
7.10 Problems............................................................143
Contents ix
III FartherfromKansas ....................................... 147
8 Inversion 149
8.1 CirclesareLines.....................................................149
8.2 WhereDoClinesGo?.................................................151
8.3 AnExamplefromtheUSAMO........................................154
8.4 OverlaysandOrthogonalCircles.......................................156
8.5 MoreOverlays.......................................................159
8.6 TheInversionDistanceFormula.......................................160
8.7 MoreExampleProblems..............................................160
8.8 WhentoInvert.......................................................165
8.9 Problems............................................................165
9 ProjectiveGeometry 169
9.1 CompletingthePlane.................................................169
9.2 CrossRatios.........................................................170
9.3 HarmonicBundles ................................................... 173
9.4 ApollonianCircles...................................................176
9.5 Poles/PolarsandBrocard’sTheorem...................................178
9.6 Pascal’sTheorem.....................................................181
9.7 ProjectiveTransformations............................................183
9.8 Examples............................................................185
9.9 Problems............................................................190
10 CompleteQuadrilaterals 195
10.1 SpiralSimilarity ..................................................... 196
10.2 Miquel’sTheorem....................................................198
10.3 TheGauss-BodenmillerTheorem......................................198
10.4 MorePropertiesofGeneralMiquelPoints..............................200
10.5 MiquelPointsofCyclicQuadrilaterals.................................201
10.6 ExampleProblems...................................................202
10.7 Problems............................................................205
11 PersonalFavorites 209
IV Appendices.................................................213
AppendixA:AnOunceofLinearAlgebra 215
A.1 MatricesandDeterminants............................................215
A.2 Cramer’sRule ....................................................... 217
A.3 VectorsandtheDotProduct...........................................217
AppendixB:Hints 221
AppendixC:SelectedSolutions 241
C.1 SolutionstoChapters1–4.............................................241
C.2 SolutionstoChapters5–7.............................................251