**Problem Books**

Volume: 27;
2016;
311 pp;
Softcover

Print ISBN: 978-0-88385-839-4

Product Code: PRB/27

List Price: $60.00

AMS Member Price: $45.00

MAA member Price: $45.00

**Electronic ISBN: 978-1-61444-411-4
Product Code: PRB/27.E**

List Price: $60.00

AMS Member Price: $45.00

MAA member Price: $45.00

# Euclidean Geometry in Mathematical Olympiads

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*Evan Chen*

MAA Press: An Imprint of the American Mathematical Society

This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class.

#### Reviews & Endorsements

The book is divided into four parts. Part I ("Fundamentals") discusses a number of basic ideas that will be used repeatedly in the sequel. I hesitate to call this part of the book a "review," because many of the topics covered here (e.g., Ceva's theorem, the power of a point) might well be new to a student who has not taken a college course in geometry. Part II ("Analytic Techniques") does not, its name notwithstanding, involve analysis, but does cover a variety of useful techniques for tackling geometric problems: computational formulas, complex numbers, and barycentric coordinates. Part III ("Further from Kansas") brings in more advanced ideas, with chapters on inversion with respect to a circle, the extended Euclidean plane (projective geometry), and complete quadrilaterals. Part IV contains a series of appendices, mostly consisting of hints and/or solutions to some of the problems in the earlier parts. A good understanding of high school geometry, and a fondness for solving problems, should be sufficient background for this book. There are topics covered here that are not generally covered in a high school course, but definitions are provided for these. The heart of a book like this is, of course, the problems. As I noted earlier, there are a great many of them, and by and large, they struck me as very difficult and involved. Even the diagrams for some of them can be a bit daunting. They should provide a good challenge for prospective test-takers, though the large number of unsolved problems might prove frustrating for some. Even if not used as the text for a geometry course, an instructor of such a course might want to keep the book handy as a potential source of challenging problems. And, as previously noted, students preparing for mathematics competitions, and their faculty coaches, should find this book very valuable.

-- Mark Hunacek, MAA Reviews

# Table of Contents

## Euclidean Geometry in Mathematical Olympiads

- Cover cov11
- Half title i2
- Copyright ii3
- Title iii4
- Series iv5
- Dedication vi7
- Contents vii8
- Preface xi12
- Preliminaries xiii14
- I Fundamentals 118
- 1 Angle Chasing 320
- 2 Circles 2340
- 3 Lengths and Ratios 4360
- 4 Assorted Configurations 5976
- 4.1 Simson Lines Revisited 5976
- 4.2 Incircles and Excircles 6077
- 4.3 Midpoints of Altitudes 6279
- 4.4 Even More Incircle and Incenter Configurations 6380
- 4.5 Isogonal and Isotomic Conjugates 6380
- 4.6 Symmedians 6481
- 4.7 Circles Inscribed in Segments 6683
- 4.8 Mixtilinear Incircles 6885
- 4.9 Problems 7087

- II Analytic Techniques 7390
- 5 Computational Geometry 7592
- 6 Complex Numbers 95112
- 6.1 What is a Complex Number? 95112
- 6.2 Adding and Multiplying Complex Numbers 96113
- 6.3 Collinearity and Perpendicularity 99116
- 6.4 The Unit Circle 100117
- 6.5 Useful Formulas 103120
- 6.6 Complex Incenter and Circumcenter 106123
- 6.7 Example Problems 108125
- 6.8 When (Not) to use Complex Numbers 115132
- 6.9 Problems 115132

- 7 Barycentric Coordinates 119136
- 7.1 Definitions and First Theorems 119136
- 7.2 Centers of the Triangle 122139
- 7.3 Collinearity, Concurrence, and Points at Infinity 123140
- 7.4 Displacement Vectors 126143
- 7.5 A Demonstration from the IMO Shortlist 129146
- 7.6 Conway's Notations 132149
- 7.7 Displacement Vectors, Continued 133150
- 7.8 More Examples 135152
- 7.9 When (Not) to Use Barycentric Coordinates 142159
- 7.10 Problems 143160

- III Farther from Kansas 147164
- IV Appendices 213230
- Bibliography 305322
- Index 307324
- About the Author 311328