**MSRI Mathematical Circles Library**

Volume: 18;
2016;
214 pp;
Softcover

MSC: Primary 51;

Print ISBN: 978-1-4704-1921-9

Product Code: MCL/18

List Price: $25.00

Individual Price: $18.75

**Electronic ISBN: 978-1-4704-3011-5
Product Code: MCL/18.E**

List Price: $25.00

Individual Member Price: $18.75

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# Geometry in Problems

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*Alexander Shen*

Translated from Russian by Marie Brodsky with assistance from Gina Yatsenko, Ray Droujkov, and Artem Astapchuk.

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

Classical Euclidean geometry, with all its triangles, circles, and
inscribed angles, remains an excellent playground for high-school
mathematics students, even if it looks outdated from the professional
mathematician's viewpoint. It provides an excellent choice of elegant
and natural problems that can be used in a course based on problem
solving.

The book contains more than 750 (mostly) easy but nontrivial
problems in all areas of plane geometry and solutions for most of
them, as well as additional problems for self-study (some with
hints). Each chapter also provides concise reminders of basic notions
used in the chapter, so the book is almost self-contained (although a
good textbook and competent teacher are always recommended). More than
450 figures illustrate the problems and their solutions.

The book can be used by motivated high-school students, as well as
their teachers and parents. After solving the problems in the book the
student will have mastered the main notions and methods of plane
geometry and, hopefully, will have had fun in the process.

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.

What a joy! Shen's “Geometry in Problems” is a gift to the school teaching world. Beautifully organized by content topic, Shen has collated a vast collection of fresh, innovative, and highly classroom-relevant questions, problems, and challenges sure to enliven the minds and clever thinking of all those studying Euclidean geometry for the first time.

This book is a spectacular resource for educators and students alike. Users will not only sharpen their mathematical understanding of specific topics but will also sharpen their problem-solving wits and come to truly own the mathematics explored. Also, Math Circle leaders can draw much inspiration for session ideas from the material presented in this book.

—James Tanton, Mathematician-at-Large, Mathematical Association of America

We learn mathematics best by doing mathematics. The author of this book recognizes this principle. He invites the reader to participate in learning plane geometry through carefully chosen problems, with brief explanations leading to much activity. The problems in the book are sometimes deep and subtle: almost everyone can do some of them, and almost no one can do all. The reader comes away with a view of geometry refreshed by experience.

—Mark Saul, Director of Competitions, Mathematical Association of America

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

#### Table of Contents

# Table of Contents

## Geometry in Problems

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface to the English Edition ix10
- Preface xi12
- Chapter 1. Measuring Line Segments 114
- Chapter 2. Measuring Angles 518
- Chapter 3. The Triangle Inequality 1124
- Chapter 4. Congruent Figures 1730
- Chapter 5. Triangle Congruence Tests 1932
- Chapter 6. Isosceles Triangles 2538
- Chapter 7. Circle 3144
- Chapter 8. Straightedge and Compass Constructions 3548
- Chapter 9. Parallel Lines 3952
- Chapter 10. Right Triangles 4962
- Chapter 11. Parallelograms 5366
- Chapter 12. Rectangle, Rhombus, Square 5770
- Chapter 13. Graph Paper 6376
- Chapter 14. Equilateral Triangles 6982
- Chapter 15. Midsegment of a Triangle 7386
- Chapter 16. Intercept Theorem 7790
- Chapter 17. Trapezoid 8396
- Chapter 18. Simple Inequalities 87100
- Chapter 19. Reflection Symmetry 91104
- Chapter 20. Central Symmetry 101114
- Chapter 21. Angles in a Circle 105118
- Chapter 22. Tangents 113126
- Chapter 23. Two Circles 119132
- Chapter 24. Circumscribed Circle and Perpendicular Bisectors 125138
- Chapter 25. Inscribed Circle (Incircle). Bisectors 129142
- Chapter 26. Inscribed and Circumscribed Quadrilaterals 133146
- Chapter 27. Area 139152
- Chapter 28. The Pythagorean Theorem 153166
- Chapter 29. Similarity 163176
- Chapter 30. Coordinates on a Line 177190
- Chapter 31. Coordinates on a Plane 183196
- Chapter 32. Common Measure 189202
- Chapter 33. Trigonometry 199212
- Afterword 209222
- Back Cover Back Cover1229

#### Readership

Students, teachers, math circles organizers, parents, and others interested in teaching and learning basic plane geometry.