Euclidean Geometry in Mathematical Olympiads

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.

There is little or nothing in this text on constructions with straightedge and compass, but that may be because this topic doesn’t much appear in mathematics competitions. There is also no chapter devoted to isometries of the plane, but rotations and reflections are occasionally mentioned in the text.

Each of the first three parts of the book contains three or four chapters, and each chapter is roughly organized the same way: a discussion of the underlying geometric theory, some worked out examples, and a collection of problems, some of them solved in part IV. The ratio of unsolved to solved problems is fairly high; I would guess roughly 5:1 or 6:1 on average. However, many of the unsolved problems are accompanied by hints, and sources are given for the many problems that come from previous competitions, solutions to which can frequently be found online. For example, solutions to the various International Mathematical Olympiad problems can be found at http://www.artofproblemsolving.com/wiki/index.php?title=IMO_Problems_and_Solutions

and solutions to the USA Mathematical Olympiad exams are here:

https://www.artofproblemsolving.com/wiki/index.php?title=USAMO_Problems_and_Solutions.

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 contest-takers, though the large number of unsolved problems might prove frustrating for some.

Because parts I through III of the text discuss a number of topics that might be part of the syllabus of an undergraduate “advanced Euclidean geometry” course (e.g., triangle centers and other associated geometric objects such as the Euler line; the nine-point circle; the theorems of Ceva and Menelaus; the complex number approach to geometry; barycentric coordinates; inversion in a circle; the extended Euclidean plane), it is natural to ask whether this book could be used as a text for such a course. I think its use for such a purpose would be problematic, for several reasons. First, the theoretical development of the material is often done in a fairly cursory manner. (Example: there is no real explanation as to why, relative to a fixed triangle, barycentric coordinates for a point exist, even though such an explanation could be given in a line or two using basic linear algebra.) Second, the author may at times be guilty of underestimating the extent to which an average undergraduate student requires hand-holding and detailed explanations; some explanations struck me as being on the terse side. For an intended audience of mathematical problem-solvers, this shouldn’t really pose much of a problem, but it could for a classroom full of undergraduates of varying abilities. Third, the average student in such a course could not begin to do most or all of the problems in the book.

Of course, 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 (mhunacek@iastate.edu) teaches mathematics at Iowa State University.