This book is an outgrowth of five years of participating in mathematical olympiads, where geometry flourishes in great vigor. The ideas, techniques, and proofs come from countless resources—lectures at MOP resources found online, discussions on the Art of Problem Solving site, or even just late-night chats with friends. The problems are taken from contests around the world, many of which I personally solved during the contest, and even a couple of which are my own creations.

As I have learned from these olympiads, mathematical learning is not passive—the only way to learn mathematics is by doing. Hence this book is centered heavily around solving problems, making it especially suitable for students preparing for national or international olympiads. Each chapter contains both examples and practice problems, ranging from easy exercises to true challenges.

Indeed, I was inspired to write this book because as a contestant I did not find any resources I particularly liked. Some books were rich in theory but contained few challenging problems for me to practice on. Other resources I found consisted of hundreds of problems, loosely sorted in topics as broad as “collinearity and concurrence”, and lacking any exposition on how a reader should come up with the solutions in the first place. I have thus written this book keeping these issues in mind, and I hope that the structure of the book reflects this.

I am indebted to many people for the materialization of this text. First and foremost, I thank Paul Zeitz for the careful advice he provided that led me to eventually publish this book. I am also deeply indebted to Chris Jeuell and Sam Korsky whose careful readings of the manuscript led to hundreds of revisions and caught errors. Thanks guys!

I also warmly thank the many other individuals who made suggestions and comments on early drafts. In particular, I would like to thank Ray Li, Qing Huang, and Girish Venkat for their substantial contributions, as well as Jingyi Zhao, Cindy Zhang, and Tyler Zhu, among many others. Of course any remaining errors were produced by me and I accept sole responsibility for them. Another special thanks also to the Art of Problem Solving fora, from which countless problems in this text were discovered and shared.