Solving Problems in Geometry

Problem books in mathematics, with the goal of training readers for Olympiads and other contests, seem to be proliferating like kudzu, to the point where, to my mind anyway, they can often be difficult to tell apart. The book under review, however, seems to stand out a bit from the pack: I don’t have any real involvement in Iowa State University’s contest training endeavors, but I do teach a two-semester sequence in upper-level geometry, and this book seems to be useful for the latter as well as for the former.

There are six chapters in the book, the first four of which are organized along the lines of substantive plane Euclidean geometry. Topics discussed in these chapters include congruence and similarity, the theorems of Ceva and Menelaus, and topics connected with circles (e.g., triangle centers and the nine-point circle, the radical axis, Ptolemy’s theorem). Within each chapter, relevant background material is developed as needed, worked-out problems are done in the text, and there are a number of end-of-chapter exercises (solutions to which appear later in the book). In many problem books, the problems can be gimmicky and very hard, thereby making them useful for contest preparation but perhaps not so useful as a source of test (or even homework) questions in an actual class. Though this is true of some of the problems that appear in these first four chapters, there are a number that are short and simple enough to function as good classroom problems.

Of course, this is not to say that the book lacks value as a contest-preparation text. Throughout the text, the authors emphasize not just the solution to a particular problem, but also make a serious effort to provide insight into how the solution might be obtained. Indeed, most of the solutions presented in the text are preceded by a discussion (sometimes longer than the solution itself!) as to how that solution might be arrived at.

In addition, the final two chapters of the book are directly geared to problem-solving. The first of these chapters begins with a summary of major results in geometry (many taken from the first four chapters), followed by two sections discussing useful techniques for solving geometric problems. (The problems discussed here are, naturally, arranged by technique rather than substantive subject matter.) In the second of these two chapters, old competition problems are discussed. Many are worked out in the text (with references to the competitions from which they are drawn) and still others are left as exercises. As in the first four chapters, solutions to the exercises are provided.

Of course, one does not have to be a contest entrant in order to appreciate a good discussion of geometric problem-solving. Anyone teaching an undergraduate course in geometry, and who can spare a lecture or two, might find these chapters useful as a general survey of solving geometric problems.

I do have a few quibbles, the first one concerning some terminology used in the book. For example, the authors define a “trapezium” to mean a convex quadrilateral with exactly one pair of parallel opposite sides. (So, for example, a parallelogram is not a trapezium.) The more common term “trapezoid” seems not to be used in the book, and it is unclear whether the authors intend these terms to be synonymous. If so, that seems to me to be a problem, similar to saying that a square is not a rectangle.

In addition, the authors use the word “inverse” to mean what I (and everybody else I know) would call the “converse”. This is a more troublesome problem than the trapezium/trapezoid issue, because I was brought up to believe that the converse and inverse of a conditional statement are two separate and very different things.

Finally, my biggest problem with the book is, unfortunately, something I have had occasion to complain about frequently in connection with other books: there is no index at all. For this reason alone, I would refuse to use the book in connection with any class I was teaching.

These issues notwithstanding, I think that this book is a useful reference for faculty members involved in contest preparation or teaching Euclidean geometry at the college level.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.