One of the things I’ve noticed about retirement is that occasionally I will see a recently published textbook and immediately think to myself that I wish it had been published several years before, back when I was teaching a course for which the book would have been useful. Because I spent each of my last seven years at Iowa State University teaching a two-semester sequence of geometry courses (Euclidean in the fall, non-Euclidean in the spring), I had that feeling more or less constantly while reading this new book by Boris Pritsker, the prolific author of such other books on geometry and problem solving as Geometrical Kaleidoscope (the first edition of which I very favorably reviewed in this column); The Equations World; Mathematical Labyrinths. Pathfinding, and Expanding Mathematical Toolbox: Interweaving Topics, Problems and Solutions.

The title of this book notwithstanding, this book is not, I think, a stand-alone text from which a student can learn undergraduate-level Euclidean geometry. Pritsker clearly assumes that the reader has encountered geometric ideas elsewhere: terms like “isosceles trapezoid” are used in the text as early as page 2, for example, and although there is an appendix covering basic terminology and theorems of Euclidean geometry, that appendix is no substitute for having encountered these ideas in a course or other textbook (perhaps one that is run concurrently with reading this book).

What this book is, is a splendid collection of fully-solved problems in geometry (Euclidean only), a wonderful resource for instructors of such a course or for highly-motivated students who wish to supplement geometric knowledge obtained elsewhere. The problems, which vary broadly in difficulty, are (except for the last chapter) arranged in roughly increasing order of difficulty, from very routine ones to much harder problems, some of them contest-level. Some of the problems, marked with an asterisk, are actually well-known results, such as Ceva’s theorem. (Side note: I was puzzled by the fact that Pritsker didn’t also mention Menelaus’s theorem, which is usually discussed in tandem with Ceva’s.)

The book is divided into two parts, “Problems” and “Solutions.” The first part consists of seven chapters. The first chapter (“Basics and Strategies”) begins with a quick overview of problem-solving strategies, illustrated by a dozen problems with (unlike the remaining chapters) solutions provided immediately after the problem. The second chapter (“Warm- Ups”) is designed, according to the author, to refresh the student’s understanding of the basic facts of geometry. The third through sixth chapters each involve a substantive topic in Euclidean geometry: triangles, quadrilaterals and polygons, circles, and area, respectively. Each of these chapters is divided into three parts (A, B and C), reflecting the difficulty of the problems. Finally, Chapter 7 (“Mixed Challenges”) is a potpourri of fairly advanced problems covering the whole spectrum of topics discussed previously; the arrangement of problems in this chapter is not according to difficulty.

The problems are of varied type—some call for proofs, some for calculation (“what is angle X?”), some for ruler-compass constructions. An instructor should be able to find lots of problems in these pages for extra homework questions, or even for test questions, since the easiest problems in the book can be answered in just a few words (example: given certain information, are two triangles necessarily congruent?).

After the seven chapters of problems, the bulk of the book presents solutions to the problems in Chapters 2 through 7. In solving these problems, Pritsker draws not just on purely synthetic reasoning but uses a range of techniques, including analytic geometry and trigonometry. (On occasion, he even uses geometric transformations like rotations.) Also, in what I think is an excellent feature of the book, Pritsker frequently gives multiple solutions to the same problem; he believes (as do I, and, I think, most mathematicians) that students benefit from seeing different approaches to a problem.

Pritsker’s writing style is mildly idiosyncratic, perhaps because English is not his first language. For example, definite articles tend to be omitted: “The book starts with chapter devoted to discussion….” However, this does not seriously affect the clarity of writing or accessibility of the text. Students should have no trouble reading this book.

There is a three-page bibliography, but quite a lot of the books mentioned in it are written in Russian or Ukrainian and are therefore likely to be out of reach to readers in the United States.

To summarize and conclude: I mentioned earlier that this book would be valuable for instructors teaching undergraduate-level geometry or for highly motivated students. I shouldn’t, however, leave readers with the impression that this is the only prospective audience for this book. For example, I don’t fall into either group, and I’m delighted that I own this book. In fact, anybody who likes geometry and problem solving will enjoy having this book on their shelves.

Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.