Geometry

438 Pages·2020·3.037 MB·English

by  Israel M. Gelfand, Tatiana Alekseyevskaya (Gelfand)

Israel M. Gelfand Tatiana Alekseyevskaya (Gelfand)
Geometry

Israel M. Gelfand
Tatiana Alekseyevskaya (Gelfand)

Geometry
Israel M. Gelfand (Deceased) Tatiana Alekseyevskaya (Gelfand)
New Brunswick, NJ, USA Rutgers, The State University of New Jersey
Highland Park, NJ, USA
ISBN 978-1-0716-0297-3 ISBN 978-1-0716-0299-7 (eBook)
https://doi.org/10.1007/978-1-0716-0299-7

Mathematics Subject Classification (2010): 51M04, 51-01

© The Authors 2020
This work is subject to copyright. All rights are reserved by the Authors, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this
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The publisher, the authors, and the editors are safe to assume that the advice and information in this
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authors or the editors give a warranty, expressed or implied, with respect to the material contained herein
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jurisdictional claims in published maps and institutional affiliations.
Cover illustration design by T. Alekseyevskaya (Gelfand), with digital version by Tatiana I. Gelfand.
Illustrations by T. Alekseyevskaya (Gelfand), with digital versions by Lyuba Pogost and Tatiana I. Gelfand.
This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered
company Springer Science+Business Media, LLC
The registered company address is: 1 New York Plaza, New York, NY 10004, U.S.A.
Contents
SeriesPreface ix
Preface xi
Introduction xvii
I PointsandLines:ALookatProjectiveGeometry 1
1 Pointsandlines ..................................................1
2 Twolinesandanangle ...........................................10
3 Threelines ..................................................... 16
4 Fourlines.Quadrilaterals .........................................25
5 Fivelines .......................................................29
6 Projectionfromapointontoaline ................................29
7 Dualconfigurationsinprojectivegeometry .........................40
8 Desarguesconfiguration ..........................................45
9 DualDesarguesconfiguration .....................................50
10 Algebraicnotationof“computerpresentation”ofconfigurations .....56
11 Polygonsandnstraightlines .....................................59
12 Convexpolygons,convexhullofnpoints ..........................62
13 SolutionofExercise3withthehelpofaDesarguesconfiguration ....65
14 OverviewofChapterI ...........................................70
II ParallelLines:ALookatAffineGeometry 71
PartI:LinesandSegments
1 ParallelStraightLines ...........................................71
2 OperationsavailableinChapterII .................................73
3 Propertiesofparallellines ........................................74
4 Segmentslyingonparallellines ...................................76
PartII:Figures
5 Parallelograms ..................................................87
6 Triangles .......................................................99
7 Trapezoids .................................................... 107
v
vi Contents
PartIII:Operationswithfigures
8 TheMinkowskyadditionoftwofigures .......................... 113
9 Parallelprojection ..............................................115
10 Paralleltranslation .............................................120
11 Centralsymmetryontheplane .................................. 132
12 Vectors ........................................................145
13 OverviewofChapterII .........................................159
AppendixforChapterII 161
1 WhywecannotdefineequalsegmentsinChapterII ............... 161
2 Parallellines,equalsegments,andtheDesarguesconfiguration .....164
3 Arithmeticoperationswithsegments .............................170
4 Segmentsandrationalnumbers ..................................177
5 Affinecoordinatesystemsontheplane ...........................181
III Area:ALookatSymplecticGeometry 189
1 Theareaofafigure .............................................189
2 Areaofaparallelogram .........................................194
3 Areaofatriangle ...............................................205
4 Areaofatrapezoid .............................................215
5 Areaofapolygon ..............................................217
6 Moreproblemsonareas ........................................ 224
7 Howtomeasuretheareaofafigure ..............................227
8 OverivewofChapterIII .........................................230
IV Circles:ALookatEuclideanGeometry 231
PartI:IntroductiontotheCircle
1 OperationsavailableinChapterIV ...............................231
2 Comparingsegments ...........................................235
3 Angles ........................................................237
4 Operationswithfigures .........................................252
PartII:Thegeometryofthetriangleandotherfigures
5 Elementsoftriangle.Congruenttriangles .........................257
6 Constructionofatrianglefromitselements .......................259
7 Relationsbetweenelementsofatriangle ..........................263
8 Propertiesofatriangle.Particularkindsoftriangles ............... 271
9 AreainEuclideangeometry .....................................281
10 ThePythagoreantheoremanditsapplications .....................285
11 Relationsbetweenlinesandpoints ...............................296
12 Speciallinesandspecialpointsinatriangle .......................308
13 Polygons ......................................................318
14 Summaryoffactsaboutdifferentquadrilaterals ................... 324
Contents vii
15 Similarity .....................................................333
PartIII:Circles
16 Circlesandpoints ..............................................339
17 Circlesandlines ...............................................343
18 Twoormorecircles ............................................347
19 Circlesandangles ..............................................353
20 Acircleandatriangle ..........................................371
21 Circlesandpolygons ...........................................381
22 Circumferenceandarc ..........................................391
23 Disksandsectors ...............................................398
24 OverviewofChapterIV ........................................ 400
Glossary 403
Preface
for the series of books written by Israel Gelfand for
high-school students
Inourcenturyofrapidchanges,itisimpossibletoknow
everything. Thegoalistolearnhowtolearn. —IsraelGelfand
There are five books written by Israel Gelfand with co-authors for high-
school students: The Method of Coordinates, Functions and Graphs,
Algebra,Trigonometry,andGeometry.
Israel Gelfand was internationally known for his teaching skills and his
remarkable ability to explain mathematical notions and concepts in an in-
teresting, “fresh,” and easy-to-understand way to a varied group of people:
from little kids, people who were not familiar with the subject at all, to
students and specialists in the field. He could amazingly feel the level of
thelistenerandadjusthisexplanationstothislevel.
In these books, Gelfand intended to cover the basics of mathematics
in a clear and simple format suitable for independent study, while inviting
students to learn and practice the material at their own pace. He wanted to
raise students’ interest in mathematics and their ability to understand and
learn. In his own words (from the preface in The Method of Coordinates),
“The most important thing a student can get from the study of mathematics
istheattainmentofahigherintellectuallevel.”
Initially, the first two books, Functions and Graphs and The Method
of Coordinates, were written for students of the Mathematical School by
CorrespondenceorganizedbyGelfandinMoscowin1964. Theywerequite
popular among students. The later books were written for the Gelfand Cor-
respondencePrograminMathematics(GCPM)1,whichGelfandestablished
atRutgersUniversityin1991andforwhichhespeciallywroteAssignments
1seehttp://www.israelmgelfand.com/egcpm.html
ix
(cid:89) SeriesPreface
with me. He wanted us to write other books—Calculus, Combinatorics,
Arithmetic,andGeometryinSpace—butunfortunately,Iwastoobusywrit-
ingtheAssignmentsandthebookGeometry,amongotherthings,andthese
bookswerenotwritten.
Gelfandviewedmathematicsasonesingleandwholeareaofknowledge,
and did not like when it was divided into separate narrow fields. Gelfand
could often find unexpected connections between different areas of math-
ematics. In his talk entitled “The Unity of Mathematics,” which he gave
during the international conference in honor of his 90th birthday, he said
“From my point of view, mathematics is a part of our culture like music,
poetryandphilosophy.”
Wehopethatthisseriesofbookswillhelpyoulearnthebasicsofmathe-
maticsfromdifferentapproaches,andmaybehelpyoudiscoverforyourself
someoftheirconnectionsandtheUnityofMathematics.
TatianaAlekseyevskaya(Gelfand)
Preface
Dearreader,
Geometryisthefifthandfinalbookintheserieswrittenforhigh-school
studentsbyIsraelGelfandwithhiscolleagues.
Whatisspecialaboutthisbook? Whyandforwhomwasitwritten?
Thisbookpresentsgeometryinanunusualway. Insteadoffocusingonlogic
and axioms, it focuses on geometrical constructions and presents concepts
inavisualform. Itstartsbyintroducingafewsimplenotionsandthengrad-
ually builds upon them. Students are invited to draw figures and “move”
them on the plane. We also introduce transformations—you can see them
illustratedonthecover.
Israel Gelfand believed that geometry is the simplest model of spatial
relationships in the world. Studying geometry will help students visualize
objects and shapes on the plane and in space, and help them develop an
intuitive understanding about how they change if they are moved. Rather
than make students memorize theorems and practice logic, Gelfand wanted
to raise students’ interest in the subject and teach them skills such as geo-
metrical vision, imagination, and creativity. These skills are very important
ineverydaylifenomatterwhatfuturepathastudentwillchoose.
Manybooksarewrittenoncalculus,butonlyarelativefewongeometry.
You can read more about the importance of geometry in our lives and a de-
scriptionofthestructureofthisbookintheIntroduction,whichispresented
inwordsascloseaspossibletoGelfand’sown.
Alltheabovemakesthisbooksuitableforawideaudience,fromstudents
withdifferentbackgrounds,toreaderswithavarietyofinterests,toeducators
andtomathematicians,whocanappreciatethisnewwayofpresentingplane
geometryinasimpleformwhileadheringtoitsdepthandrigor.
In Gelfand’s approach, pictures play an essential role. He called the
pictures “the main beauty of the book” which would distinguish this book
xi