Episodes in ninteenth and twentieth century Euclidean geometry

189 Pages·1996·0.988 MB·English

by  Honsberger, Ross

Episodes in Nineteenth and
Twentieth Century Geometry
Prof. Honsberger has succeeded in fi nding unex-
pected and little known properties of such funda-
E
mental fi gures as triangles, results that deserve to p
is Episodes in
be better known. He has laid the foundations for o
d
e
his proofs with almost entirely synthetic methods s
i Nineteenth and
easily accessible to students of Euclidean geome- n
N
try early on. While in most of his other books Hons- i
n
e Twentieth Century
berger presents each of his gems, morsels, and t
e
plums as self-contained tidbits, in this volume he e
n
t Euclidean Geometry
connects chapters with some deductive threads. h
a
He includes exercises and gives their solutions at n
d
the end of the book. T
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n
t
i
e
t
h
Ross Honsberger was born in Toronto, Canada, C
e
and attended the University of Toronto. After n
t
u
more than a decade of teaching secondary school r
y
mathematics in Toronto, he took advantage of E
u
c
a sabbatical to study at the University of Water- l
i
d
loo, Canada. He joined its faculty in 1964 in the e
a
n
Department of Combinatorics and Optimization
G
and is now Professor Emeritus. He has published e
o
m
13 best-selling books with the MAA including
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t
Mathematical Diamonds, Mathematical Chestnuts r
y
from Around the World, From Erdös to Kiev, and
Ingenuity in Mathematics.
H
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ISBN 978-0-88385-639-0 s
b Ross Honsberger
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9 780883 856390
Anneli Lax New Mathematical Library | Vol. 37
EPISODES IN NINETEENTH
AND TWENTIETH CENTURY
EUCLIDEAN GEOMETRY
NEWMATHEMATICALLIBRARY
publishedby
TheMathematicalAssociationofAmerica
EditorialCommittee
PaulZorn,Chair AnneliLax
StOlafCollege NewYorkUniversity
JudithE.Broadwin JerichoHighSchool
UnderwoodDudley DePauwUniversity
EdwardM.Harris
DavidSanchez SanAntonioCollege
MichaelJ.McAsey BradleyUniversity
PeterUngar
The New MathematicalLibrary(NML)was begunin 1961by the School
MathematicsStudyGrouptomakeavailabletohighschoolstudentsshort
expositorybooksonvarioustopicsnotusuallycoveredinthehighschool
syllabus. In three decades the NML has matured into a steadily growing
series of some thirty titles of interest not only to the originally intended
audience,buttocollegestudentsandteachersatalllevels.Previouslypub-
lishedbyRandomHouseandL.W.Singer,theNMLbecameapublication
seriesoftheMathematicalAssociationofAmerica(MAA)in1975.Under
theauspicesoftheMAAtheNMLwillcontinuetogrowandwillremain
dedicatedtoitsoriginalandexpandedpurposes.
EPISODES IN NINETEENTH
AND TWENTIETH CENTURY
EUCLIDEAN GEOMETRY
by
Ross Honsberger
UniversityofWaterloo
37
THEMATHEMATICALASSOCIATIONOFAMERICA
©Copyright 1995 by The Mathematical Association of America (Inc.)
All rights reserved under International
and Pan-American Copyright Conventions.
Published in Washington, D.C. by
The Mathematical Association of America
Library of Congress Catalog Card Number: 94-079528
Print ISBN 978-0-88385-639-0
Electronic ISBN 978-0-88385-951-3
Manufactured in the United States of America
Note to the Reader
Thisbookisoneofaserieswrittenbyprofessionalmathematiciansinorder
tomakesomeimportantmathematicalideasinterestingandunderstandabletoa
largeaudienceofhighschoolstudentsandlaymen.Mostofthevolumesinthe
NewMathematicalLibrarycovertopicsnotusuallyincludedinthehighschool
curriculum; they vary in difficulty, and, even within a single book, some parts
requireagreaterdegreeofconcentrationthanothers.Thus,whileyouneedlittle
technicalknowledgetounderstandmostofthesebooks,youwillhavetomake
anintellectualeffort.
If you have so far encountered mathematics only in classroom work, you
should keep in mind that a book on mathematics cannot be read quickly. Nor
mustyouexpecttounderstandallpartsofthebookonfirstreading.Youshould
feel free to skip complicated parts and return to them later; often an argument
willbeclarifiedbyasubsequentremark.Ontheotherhand,sectionscontaining
thoroughlyfamiliarmaterialmaybereadveryquickly.
The best way to learn mathematics is to do mathematics, and each book
includes problems some of which may require considerable thought. You are
urgedtoacquirethehabitofreadingwithpaperandpencilinhand;inthisway,
mathematicswillbecomeincreasinglymeaningfultoyou.
Theauthorsandeditorialcommitteeareinterestedinreactionstothebooksin
thisseriesandhopethatyouwillwriteto:AnneliLax,Editor,NewMathematical
Library,NewYorkUniversity,TheCourantInstituteofMathematicalSciences,
251MercerStreet,NewYork,N.Y.10012.
TheEditors
NEWMATHEMATICALLIBRARY
1. Numbers:RationalandIrrationalbyIvanNiven
2. WhatisCalculusAbout?byW.W.Sawyer
3. AnIntroductiontoInequalitiesbyE.F.BeckenbachandR.Bellman
4. GeometricInequalitiesbyN.D.Kazarinoff
5. TheContestProblemBookIAnnualh.s.math.exams,1950–1960.Compiledandwith
solutionsbyCharlesT.Salkind
6. TheLoreofLargeNumbersbyP.J.Davis
7. UsesofInfinitybyLeoZippin
8. GeometricTransformationsIbyI.M.Yaglom,translatedbyA.Shields
9. ContinuedFractionsbyCarlD.Olds
10. (cid:2)ReplacedbyNML-34
11. HungarianProblemBooksIandII, BasedontheEo¨tvo¨s
12. Competitions1894–1905and1906–1928,translatedbyE.Rapaport
13. EpisodesfromtheEarlyHistoryofMathematicsbyA.Aaboe
14. GroupsandTheirGraphsbyE.GrossmanandW.Magnus
15. TheMathematicsofChoicebyIvanNiven
16. FromPythagorastoEinsteinbyK.O.Friedrichs
17. TheContestProblemBookIIAnnualh.s.math.exams1961–1965.Compiledandwith
solutionsbyCharlesT.Salkind
18. FirstConceptsofTopologybyW.G.ChinnandN.E.Steenrod
19. GeometryRevisitedbyH.S.M.CoxeterandS.L.Greitzer
20. InvitationtoNumberTheorybyOysteinOre
21. GeometricTransformationsIIbyI.M.Yaglom,translatedbyA.Shields
22. ElementaryCryptanalysis—AMathematicalApproachbyA.Sinkov
23. IngenuityinMathematicsbyRossHonsberger
24. GeometricTransformationsIIIbyI.M.Yaglom,translatedbyA.Shenitzer
25. TheContestProblemBookIIIAnnualh.s.math.exams.1966–1972.Compiledandwith
solutionsbyC.T.SalkindandJ.M.Earl
26. MathematicalMethodsinSciencebyGeorgePolya
27. InternationalMathematicalOlympiads1959–1977.Compiledandwithsolutions
byS.L.Greitzer
28. TheMathematicsofGamesandGamblingbyEdwardW.Packel
29. TheContestProblemBookIVAnnualh.s.math.exams. 1973–1982.Compiledandwith
solutionsbyR.A.Artino,A.M.Gaglione,andN.Shell
30. TheRoleofMathematicsinSciencebyM.M.SchifferandL.Bowden
31. InternationalMathematicalOlympiads1978–1985andfortysupplementaryproblems.
CompiledandwithsolutionsbyMurrayS.Klamkin
32. RiddlesoftheSphinxbyMartinGardner
33. U.S.A.MathematicalOlympiads1972–1986.Compiledandwithsolutions
byMurrayS.Klamkin
34. GraphsandTheirUsesbyOysteinOre.RevisedandupdatedbyRobinJ.Wilson
35. ExploringMathematicswithYourComputerbyArthurEngel
36. GameTheoryandStrategybyPhilipStraffin
37. EpisodesinNineteenthandTwentiethCenturyEuclideanGeometry
byRossHonsberger
Othertitlesinpreparation.
Contents
Preface...................................................................ix
Introduction..............................................................xi
1. CleaversandSplitters..................................................1
2. TheOrthocenter......................................................17
3. OnTriangles.........................................................27
4. OnQuadrilaterals....................................................35
5. APropertyofTriangles...............................................43
6. TheFuhrmannCircle.................................................49
7. TheSymmedianPoint................................................53
8. TheMiquelTheorem.................................................79
9. TheTuckerCircles...................................................87
10. TheBrocardPoints...................................................99
11. TheOrthopole......................................................125
12. OnCevians.........................................................137
13. TheTheoremofMenelaus...........................................147
SuggestedReading......................................................155
SolutionstotheExercises................................................157
Index...................................................................173

Preface
It isalwaysgratifyingto discover that itiswithin one’sabilitytoappreciate
a mathematics book and to read it with pleasure. I have often dreamt what a
joy it would be to get to know some of the elementary gems that are surely
presentineverybranchofmathematics,onlytobedismayedbytheliteratureI
havebeenabletofind.Undoubtedlythegemsarethere,buttheyoftenlieburied
intextbooksorcomprehensivereferenceworks.Oneisfrequentlyleftwiththe
unhappy choice of undertaking a prolonged study of the field or giving up the
ideaaltogether.Whileittakesaknowledgeablescholartowritesomethingoutof
theordinary,thededicatedspecialistcangetcarriedawaywithdiscussionsthat
one comes to appreciate only after long and serious study. Unfortunately, this
makesitverydifficultforgeneralreaderstodisentangletheelementarygemsof
theirheart’sdesire.Ontopofthis,whatpassesforaproofisoftensoconciseor
sketchythatitisreadilyunderstandableonlytosomeonewhoalreadyknowsthe
subject.
Iwoulddearlylovetobeabletopromisethatyouwillfindnosuchfrustrations
in the presentwork. WhatI can promiseisa collectionofessays that doesnot
attempttocoveralargeamountofmaterial,andthateachtopichasbeenextricated
fromthemassofmaterialinwhichitisusuallyfoundandgivenaselementary
andfullatreatmentasisreasonablypossible.Thereisnosensepretendingthere
isanywayaroundtheneedtolayfoundationsforone’sproofs,butbyselecting
fromthemostaccessibletopicsIhopethatmanyreaderswillbeabletodelight
inthesegemswithaminimumofpreliminaries.
With one exception, everything that is beyond a high school background is
developed as the need arises; the basic properties of the nine-point circle are
covered in so many places that their proofs have not been included here. You
shouldbeadvised,however,thatderivationsgiveninoneessayarenotrepeated
inalaterone.Thus,whilesamplingfromthefirstfewessaysmightnotincura
greatlossofcontinuity,thelateressaysdomakeuseofconceptsandresultsthat
have been considered in earlier discussions, and you might have to look in an
earlieressayforexplanationsofanitemthatisencounteredlateron.
So much depends on your mood when you take up a book like this. Most
assuredly,theseessaysarenotintendedtobeburdensomestudies;onthecontrary,
althoughsome concentrationisrequiredfortheirappreciation,Ihopeyouwill
look forward to some relaxing entertainment and enjoy them as you would a
beautifulpieceofmusic.
ItisapleasuretothankPaulZornandthemembersoftheNewMathematical
Library Subcommittee, and Don Albers and Beverly Ruedi, for the excellence
oftheireditorialworkandguidanceofthebookthroughpublication.Particular
thanksaredueProfessorsDonChakerian,UniversityofCaliforniaatDavis,and
Basil Gordon,UniversityofCaliforniaatLosAngeles,fortheirreviewsofthe