The Structure of This Book

Loosely, each of the chapters is divided into the following parts.

1. • A theoretical portion, describing a set of related theorems and tools,

2. • One or more examples demonstrating the application of these tools, and

3. • A set of several practice problems.

The theoretical portion consists of theorems and techniques, as well as particular geometric configurations. The configurations typically reappear later on, either in the proof of another statement or in the solutions to exercises. Consequently, recognizing a given configuration is often key to solving a particular problem. We present the configurations from the same perspective as many of the problems.

The example problems demonstrate how the techniques in the chapter can be used to solve problems. I have endeavored to not merely provide the solution, but to explain how it comes from, and how a reader would think of it. Often a long commentary precedes the actual formal solution, and almost always this commentary is longer than the solution itself. The hope is to help the reader gain intuition and motivation, which are indispensable for problem solving.

Finally, I have provided roughly a dozen practice problems at the end of each chapter. The hints are numbered and appear in random order in Appendix B, and several of the solutions in Appendix C. I have also tried to include the sources of the problems, so that a diligent reader can find solutions online (for example on the Art of Problem Solving forums, www.aops.com). A full listing of contest acronyms appears in Appendix D.

The book is organized so that earlier chapters never require material from later chapters. However, many of the later chapters approximately commute. In particular, Part III does not rely on Part II. Also, Chapters 6 and 7 can be read in either order. Readers are encouraged to not be bureaucratic in their learning and move around as they see fit, e.g., skipping complicated sections and returning to them later, or moving quickly through familiar material.

Centers of a Triangle

Throughout the text we refer to several centers of a triangle. For your reference, we define them here.

It is not obvious that these centers exist based on these definitions; we prove this in Chapter 3. For now, you should take their existence for granted.