**MSRI Mathematical Circles Library**

Volume: 26;
2021;
177 pp;
Softcover

MSC: Primary 00; 51; 52; 14; 97;

**Print ISBN: 978-1-4704-4879-0
Product Code: MCL/26**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-6521-6
Product Code: MCL/26.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

# Mathematics via Problems: Part 2: Geometry

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*Alexey A. Zaslavsky; Mikhail B. Skopenkov*

A co-publication of the AMS and the Mathematical Sciences Research Institute

This book is a translation from Russian of
Part II of the book Mathematics Through Problems: From Olympiads
and Math Circles to Profession. Part I, Algebra, was
recently published in the same series. Part III,
Combinatorics, will be published soon.

The main goal of this book is to develop important parts of
mathematics through problems. The authors tried to put together
sequences of problems that allow high school students (and some
undergraduates) with strong interest in mathematics to discover and
recreate much of elementary mathematics and start edging into more sophisticated
topics such as projective and affine geometry, solid geometry, and so
on, thus building a bridge between standard high school exercises and
more intricate notions in geometry.

Definitions and/or references for material that is not standard in
the school curriculum are included. To help students that might be
unfamiliar with new material, problems are carefully arranged to
provide gradual introduction into each subject. Problems are often
accompanied by hints and/or complete solutions.

The book is based on classes taught by the authors at different
times at the Independent University of Moscow, at a number of Moscow
schools and math circles, and at various summer schools. It can be
used by high school students and undergraduates, their teachers, and
organizers of summer camps and math circles.

In the interest of fostering a greater awareness and appreciation
of mathematics and its connections to other disciplines and everyday
life, MSRI and the AMS are publishing books in the Mathematical
Circles Library series as a service to young people, their parents and
teachers, and the mathematics profession.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

#### Readership

High school students and their mentors and teachers.

#### Table of Contents

# Table of Contents

## Mathematics via Problems: Part 2: Geometry

- Cover Cover11
- Title page iii4
- Foreword ix10
- Introduction xiii14
- What this book is about and who it is for xiii14
- Learning by solving problems xiv15
- Parting words By A.Ya.Kanel-Belov xv16
- Olympiads and mathematics xv16
- Research problems for high school students xvi17
- How this book is organized xvi17
- Resources and literature xvi17
- Acknowledgments xvii18
- Numbering and notation xvii18
- References xix20

- Chapter 1. Triangle 122
- 1. Carnot’s principle (1) By V.Yu.Protasov and A.A.Gavrilyuk 223
- 2. The center of the inscribed circle (2) By V.Yu.Protasov 425
- 3. The Euler line By V.Yu.Protasov 728
- 4. Carnot’s formula (2*) By A.D.Blinkov 829
- 5. The orthocenter, orthotriangle, and nine-point circle (2) By V.Yu.Protasov 1132
- 6. Inequalities involving triangles (3*) By V.Yu.Protasov 1334
- 7. Bisectors, heights, and circumcircles (2) By P.A.Kozhevnikov 1536
- 8. \enquote{Semi-inscribed} circle (3*) By P.A.Kozhevnikov 1940
- 9. The generalized Napoleon’s theorem (2*) By P.A.Kozhevnikov 2546
- 10. Isogonal conjugation and the Simson line (3*) By A.V.Akopyan 3152
- Additional reading 3859

- Chapter 2. Circle 3960
- 1. The simplest properties of a circle (1) By A.D.Blinkov 3960
- 2. Inscribed angles (1) By A.D.Blinkov and D.A.Permyakov 4263
- 3. Inscribed and circumscribed circles (2) By A.A.Gavrilyuk 4667
- 4. The radical axis (2) By I.N.Shnurnikov and A.I.Zasorin 4768
- 5. Tangency (2) By I.N.Shnurnikov and A.I.Zasorin 4970
- 6. Ptolemy’s and Casey’s Theorems (3*) By A.D.Blinkov and A.A.Zaslavsky 5071

- Chapter 3. Geometric transformations 5576
- 1. Applications of transformations (1) By A.D.Blinkov 5576
- 2. Classification of isometries of the plane (2) By A.B.Skopenkov 6182
- 3. Classification of isometries of space (3*) By A.B.Skopenkov 6384
- 4. An application of similarity and homothety (1) By A.D.Blinkov 6586
- 5. Rotational homothety (2) By P.A.Kozhevnikov 7192
- 6. Similarity (1) By A.B.Skopenkov 7697
- 7. Dilation to a line (2) By A.Ya.Kanel-Belov 7798
- 8. Parallel projection and affine transformations (2) By A. B. Skopenkov 7899
- 9. Central projection and projective transformations (3) By A. B. Skopenkov 81102
- 10. Inversion (2) By A. B. Skopenkov 83104
- Additional reading 86107

- Chapter 4. Affine and projective geometry 87108
- Chapter 5. Complex numbers and geometry (3) By A.A.Zaslavsky 99120
- Chapter 6. Constructions and loci 105126
- Chapter 7. Solid geometry 125146
- 1. Drawing (2) By A.B.Skopenkov 125146
- 2. Projections (2) By M.A.Korchemkina 126147
- 3. Regular polyhedra (3) 130151
- 4. Higher-dimensional space (4*) By A.Ya.Kanel-Belov 134155
- 4.A. Simplest polyhedra in higher-dimensional space By Yu.M.Burman and A.Ya.Kanel-Belov 134155
- 4.B. Multi-dimensional volumes 138159
- 4.C. Volumes and intersections 139160
- 4.D. Research problems 140161
- 4.E. Partitions into parts of smaller diameter By A.M.Raigorodsky 141162
- Suggestions, solutions, and answers 141162

- Additional reading 143164

- Chapter 8. Miscellaneous geometry problems 145166
- Bibliography 171192
- Index 175196
- Back Cover Back Cover1200