Euclidean Geometry in Mathematical Olympiads (Maa Problem) - Evan Chen (2016)

This is a problem book in Euclidean plane geometry, written by an undergraduate at MIT with extensive experience in, and expertise at mathematical competitions and problem solving. The principal intended audience is students preparing for some kind of Olympiad or competition, and for such people this book should prove quite valuable. It is not only filled with a number of worked examples and lots of problems (some accompanied by solutions) but also contains discussions of general theory, specific solution techniques, and helpful advice as to when to, and when not to, apply certain methods.

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