Mathematics via Problems: Part 2: Geometry

200 Pages·2021·8.279 MB·English

by  Alexey A. Zaslavsky, Mikhail B. Skopenkov

Mathematical Circles Library
Mathematics via Problems
Geometry
PART 2:
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, California Providence, Rhode Island
Advisory Board for the MSRI/Mathematical Circles Library
TituAndreescu ZvezdelinaStankova
DavidAuckly JamesTanton
H´el`eneBarcelo RaviVakil
ZumingFeng DianaWhite
TonyGardiner IvanYashchenko
AndyLiu PaulZeitz
AlexanderShen JoshuaZucker
TatianaShubin(Chair)
ScientificEditor: DavidScott
This work was originally published in Russian by “MCNMO” under the title
(cid:2)pementy matematiki v zadaqah,(cid:2)c 2018. The present translationwas created
under license for the American Mathematical Society and is published by permission.
This volume is published withthe generous support of the Simons
Foundation and Tom Leighton and Bonnie Berger Leighton.
2020 Mathematics Subject Classification. Primary 00-01,
00A07, 51-01, 52-01, 14-01, 97G10,97-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mcl-26
Library of Congress Cataloging-in-Publication Data
Names: Zaslavski˘ı,Alekse˘ıAleksandrovich,1960–author. |Skopenkov,MikhailB.,1983–author.
Title: Mathematicsviaproblems. Part2. Geometry/AlexeyA.Zaslavsky,MikhailB.Skopenkov;
translatedbySergeiShubinandPaulZeitz.
Othertitles: E˙lementymatematikivzadachakh. English
Description: Berkeley, California: MSRI Mathematical Sciences Research Institute; Providence,
Rhode Island American Mathematical Society, [2021] | Series: MSRI mathematical circles
library,1944-8074;26|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2020057238|ISBN9781470448790(paperback)|9781470465216(ebook)
Subjects: LCSH:Geometry–Studyandteaching. |Problemsolving. |Geometry–Problems,exer-
cises,etc. |AMS:General–Instructionalexposition(textbooks,tutorialpapers,etc.). |Gen-
eral–Generalandmiscellaneousspecifictopics–Problembooks. |Geometry–Instructional
exposition (textbooks, tutorial papers, etc.). | Convex and discrete geometry – Instructional
exposition (textbooks, tutorial papers, etc.). | Algebraic geometry – Instructional exposition
(textbooks, tutorial papers, etc.). | Mathematics education – Geometry – Comprehensive
works. |Mathematicseducation–Instructionalexposition(textbooks,tutorialpapers,etc.).
Classification: LCCQA462.Z372021|DDC516.0071–dc23
LCrecordavailableathttps://lccn.loc.gov/2020057238
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Contents
Foreword ix
Problems, exercises, circles, and olympiads ix
Why this book, and how to use it x
English-language references xi
Introduction xiii
What this book is about and who it is for xiii
Learning by solving problems xiv
Parting words By A.Ya.Kanel-Belov xv
Olympiads and mathematics xv
Research problems for high school students xvi
How this book is organized xvi
Resources and literature xvi
Acknowledgments xvii
Numbering and notation xvii
Notation xviii
References xix
Chapter 1. Triangle 1
1. Carnot’s principle (1) By V.Yu.Protasov and A.A.Gavrilyuk 2
Suggestions, solutions, and answers 3
2. The center of the inscribed circle (2) By V.Yu.Protasov 4
Suggestions, solutions, and answers 5
3. The Euler line By V.Yu.Protasov 7
Suggestions, solutions, and answers 7
∗
4. Carnot’s formula (2 ) By A.D.Blinkov 8
Suggestions, solutions, and answers 9
5. The orthocenter, orthotriangle, and nine-point circle (2)
By V.Yu.Protasov 11
Suggestions, solutions, and answers 13
∗
6. Inequalities involving triangles (3 ) By V.Yu.Protasov 13
Suggestions, solutions, and answers 14
7. Bisectors, heights, and circumcircles (2) By P.A.Kozhevnikov 15
Suggestions, solutions, and answers 16
v
vi CONTENTS
∗
8. “Semi-inscribed” circle (3 ) By P.A.Kozhevnikov 19
Main series of problems—1 19
Main series of problems—2 20
Supplementary problems—1 20
Supplementary problems—2 20
Suggestions, solutions, and answers 21
∗
9. The generalized Napoleon’s theorem (2 )
By P.A.Kozhevnikov 25
Introductory problems 25
Formulation and proof of the generalized Napoleon’s theorem 26
Suggestions, solutions, and answers 27
∗
10. Isogonal conjugation and the Simson line (3 )
By A.V.Akopyan 31
Suggestions, solutions, and answers 33
Additional reading 38
Chapter 2. Circle 39
1. The simplest properties of a circle (1) By A.D.Blinkov 39
Suggestions, solutions, and answers 40
2. Inscribed angles (1) By A.D.Blinkov and D.A.Permyakov 42
Suggestions, solutions, and answers 44
3. Inscribed and circumscribed circles (2) By A.A.Gavrilyuk 46
Suggestions, solutions, and answers 47
4. The radical axis (2) By I.N.Shnurnikov and A.I.Zasorin 47
5. Tangency (2) By I.N.Shnurnikov and A.I.Zasorin 49
∗
6. Ptolemy’s and Casey’s Theorems (3 ) By A.D.Blinkov and
A.A.Zaslavsky 50
6.A. Ptolemy’s Theorem 50
6.B. Casey’s Theorem 51
Suggestions, solutions, and answers 52
Chapter 3. Geometric transformations 55
1. Applications of transformations (1) By A.D.Blinkov 55
Suggestions, solutions, and answers 57
2. Classification ofisometries oftheplane(2)By A.B.Skopenkov 61
Hints 63
3. Classification of isometries of space (3*) By A.B.Skopenkov 63
Hints 65
4. An application of similarity and homothety (1)
By A.D.Blinkov 65
Suggestions, solutions, and answers 67
5. Rotational homothety (2) By P.A.Kozhevnikov 71
5.A. Introductory problems involving cyclists 71
5.B. Main problems 72
5.C. Additional problems 73
CONTENTS vii
Suggestions, solutions, and answers 73
6. Similarity (1) By A.B.Skopenkov 76
7. Dilation to a line (2) By A.Ya.Kanel-Belov 77
Suggestions, solutions, and answers 78
8. Parallel projection and affine transformations (2)
By A. B. Skopenkov 78
Suggestions, solutions, and answers 80
9. Central projection and projective transformations (3)
By A. B. Skopenkov 81
10. Inversion (2) By A. B. Skopenkov 83
Additional reading 86
Chapter 4. Affine and projective geometry 87
1. Mass points (2) By A.A.Gavrilyuk 87
Suggestions, solutions, and answers 89
2. The cross-ratio (2) By A.A.Gavrilyuk 90
Suggestions, solutions, and answers 92
3. Polarity (2) By A.A.Gavrilyuk and P.A.Kozhevnikov 93
Fundamental properties and introductory problems 93
Main problems 94
Additional problems 95
Suggestions, solutions, and answers 96
Additional reading 98
Chapter 5. Complex numbers and geometry (3) By A.A.Zaslavsky 99
1. Complex numbers and elementary geometry 99
Suggestions, solutions, and answers 101
2. Complex numbers and Möbius transformations 102
Additional problems 103
Suggestions, solutions, and answers 103
Additional reading 104
Chapter 6. Constructions and loci 105
1. Loci (1) By A.D.Blinkov 105
Suggestions, solutions, and answers 106
2. Construction and loci problems involving area (1)
By A.D.Blinkov 111
Suggestions, solutions, and answers 113
3. Construction toolbox (2) By A.A.Gavrilyuk 117
Suggestions, solutions, and answers 119
∗
4. Auxiliary constructions (2 ) By I.I.Shnurnikov 120
Suggestions, solutions, and answers 121
Additional reading 123
Chapter 7. Solid geometry 125
1. Drawing (2) By A.B.Skopenkov 125
viii CONTENTS
Suggestions, solutions, and answers 126
2. Projections (2) By M.A.Korchemkina 126
2.A. Projections of figures constructed from cubes 126
2.B. Trajectories 128
3. Regular polyhedra (3) 130
3.A. Inscribed and circumscribed polyhedra
By A.Ya.Kanel-Belov 130
Suggestions, solutions, and answers 132
3.B. Symmetries
By A.B.Skopenkov 132
∗
4. Higher-dimensional space (4 ) By A.Ya.Kanel-Belov 134
4.A. Simplest polyhedra in higher-dimensional space
By Yu.M.Burman and A.Ya.Kanel-Belov 134
4.B. Multi-dimensional volumes 138
4.C. Volumes and intersections 139
4.D. Research problems 140
4.E. Partitions into parts of smaller diameter
By A.M.Raigorodsky 141
Suggestions, solutions, and answers 141
Additional reading 143
Chapter 8. Miscellaneous geometry problems 145
1. Geometric optimization problems (2) By A.D.Blinkov 145
Suggestions, solutions, and answers 147
2. Area (2) By A.D.Blinkov 151
Suggestions, solutions, and answers 152
∗
3. Conic sections (3 ) By A.V.Akopyan 158
Suggestions, solutions, and answers 161
∗
4. Curvilinear triangles and non-Euclidean geometry (3 )
By M.B.Skopenkov 166
Additional problems 168
Suggestions, solutions, and answers 169
Additional reading 169
Bibliography 171
Index 175
Foreword
Problems, exercises, circles, and olympiads
This is a translation of Part 2 of the book Mathematics via Problems by
A.B.Skopenkov, M.B.Skopenkov, and A.A.Zaslavsky, and is part of the
AMS/MSRI Mathematical Circles Library series. The goal of this series is
to build a body of works in English that help to spread the “math circle”
culture.
A mathematical circle is an eastern European notion. Math circles are
similar to what most Americans would call a math club for kids, but with
several important distinguishing features.
First, they are vertically integrated: young students may interact with
older students, college students, graduate students, industrial mathemati-
cians, professors, and even world-class researchers, all in the same room.
The circle is not so much a classroom as a gathering of young initiates with
elder tribespeople, who pass down folklore.
Second, the “curriculum,” such as it is, is dominated by problems rather
than specific mathematical topics. A problem, in contrast to an exercise,
is a mathematical question that one doesn’t know how, at least initially, to
approach. For example, “What is 3 times 5?” is an exercise for most people
but a problem for a very young child. Computing 534 is also an exercise,
conceptually very much like the first example, certai(cid:2)nly harder, but only in
a “technical” sense. And a question like “Evaluate 7e5xsin3xdx” is also
2
an exercise—for calculus students—a matter of “merely” knowing the right
algorithm and how to apply it.
Problems, by contrast, do not come with algorithms attached. By their
very nature, they require investigation, which is both an art and a science,
demandingtechnicalskillalongwithfocus,tenacity,andinventiveness. Math
circles teach students these skills, not with formal instruction, but by having
themdo math andobserveothersdoingmath. Studentslearnthataproblem
worthsolvingmayrequirenotminutesbutpossiblyhours,days,orevenyears
of effort. They work on some of the classic folklore problems and discover
how these problems can help them investigate other problems. They learn
how not to give up and how to turn errors or failures into opportunities for
more investigation. A child in a math circle learns to do exactly what a
ix