You are here

Geometry in Problems

Alexander Shen
Publication Date: 
Number of Pages: 
MSRI Mathematical Circles Library
Problem Book
[Reviewed by
Mark Hunacek
, on

This book, an English translation of a Russian text published in 2013, teaches elementary plane Euclidean geometry by means of numerous problems. Unlike other geometry problem books, such as Chen’s Euclidean Geometry in Mathematical Olympiads and Grigorieva’s Methods of Solving Complex Geometry Problems, the problems in this book are (for the most part, anyway) fairly simple, simple enough to assign as homework problems or even exam questions. Chen and Grigorieva are training students for Olympiads or other competitions; Shen is teaching basic geometry.

The intended audience, according to the back cover, consists of “motivated high-school students, as well as their teachers and parents”; consistent with this, the topic coverage is broad but not deep. The book is concerned with the elementary (and strictly Euclidean) geometry of triangles, circles, parallelograms, and trapezoids. Some deeper results are alluded to, but not developed in the text itself. The approach is generally intuitive-synthetic (i.e., proofs, but without a rigorous axiomatic development), but a few chapters at the end discuss analytic geometry and trigonometry. Geometric transformations are briefly referred to (there is a chapter on reflection symmetry and another on centers of symmetry of geometric figures) but there is no real systematic development of the theory of isometries in the plane. The penultimate chapter of the book is on commensurable segments and touches on number theoretical issues such as the greatest common divisor and the Euclidean algorithm.

Constructions with straightedge and compass are discussed throughout the text, but the focus is on teaching standard constructions; there is little attention paid to the really interesting question of what constructions are possible.

This material is spread out over 33 chapters, very few of them more than ten pages long and all having the same organizational structure. Each begins with a series of problems for which most solutions are immediately provided, and then there is a section (called Additional Problems) consisting of unsolved homework problems. Occasionally, basic definitions are inserted between the problems.

Since solutions to the first set of problems in each chapter are provided immediately after the statement of the problem, this isn’t really an “inquiry based learning” kind of book, however. Many results that in a standard geometry book would appear as a theorem and accompanying proof here appear as a problem and accompanying solution. The solved problems, however, go well beyond these standard theorems and include computations and other results that make for good exercises. The total number of problems, both solved and unsolved, is 784.

As noted above, some important and nontrivial results appear among the Additional Problems. For example, Ceva’s theorem appears, as does Bolyai’s theorem that if two polygons have the same area, then one can be chopped up and reassembled to form the other. Another problem describes Archimedes’ “neusis” construction (though that term isn’t used) for trisecting an angle with a compass and notched straightedge, and asks the reader to prove that it works.

In this country, there is apparently a current (and, I think, lamentable) trend to de-emphasize proofs in most high school geometry classes. For this reason, and because the book lacks the trappings of a typical high school text (multicolored pictures, boxed results, etc.), it probably won’t find much use as a text in a typical high school here. Moreover, in view of the basically elementary topic coverage, this book isn’t really suitable as a text for a typical upper-level college course in Euclidean geometry either. Perhaps the best use of this book would be as a text for a lower-level college geometry course, especially one designed primarily for teacher training. Also, the large assortment of problems might well provide a useful resource for instructors.

Because this text is translated from the Russian, the English is occasionally idiosyncratic (e.g., the word “height” is used instead of “altitude”, thus leading, for example, to statements involving the intersection of the heights of a triangle). This isn’t a serious problem, however. What is a serious problem, though, is the absence of any Index. I have, unfortunately, noted this problem in a couple of other books recently. There should be a law prohibiting this.

Buy Now

Mark Hunacek ( teaches mathematics at Iowa State University. 

See the table of contents in the publisher's webpage.