# 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

… A good understanding of high school geometry, and a fondness for solving problems, should be sufficient background for this book. … 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