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Lemmas in Olympiad Geometry

Titu Andreescu, Sam Korsky, and Cosmin Pohoata
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Mark Hunacek
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Problem books in mathematics, and particularly problem books designed to help prepare students for mathematics competitions, seem to be quite popular nowadays, so much so that entire series of them exist. The book under review is one of 30 published by XYZ Press, which, as was pointed out in our review of 111 Problems in Algebra and Number Theory, “is the publishing arm of Awesome Math, a training company founded in 2006 by well-known Olympiad coach Titu Andreescu. They put out a large number of similar problem books that are distributed through the AMS bookstore.”

This book, devoted to geometry, competes with other recent titles such as Euclidean Geometry in Mathematical Olympiads by Chen, and Grigorieva’s Methods of Solving Complex Geometry Problems, just to name two books along the same lines that are, courtesy of this column, now on my shelf. A few minutes spent on the internet will disclose quite a number of other books addressing geometry problems at competition level. This book holds up quite well against this competition.

I am not involved in my university’s mathematics problem-solving program, but I do teach a two-semester upper-level course in geometry (Euclidean the first semester, non-Euclidean the next) and am therefore always on the lookout for books that might contain interesting problems for use in class or as assignments. I particularly liked this book, because it seemed to me to have somewhat more of a “textbook feel” to it than do many books of this sort, and has a pretty good ratio (roughly 50:50) of solved to unsolved problems.

The book is divided into 25 chapters, organized by topics in Euclidean geometry. (Non-Euclidean geometry is not covered, though projective geometry is occasionally hinted at.) The topics covered are those that might appear in a course in advanced Euclidean geometry at the college level, including: the power of a point, various objects associated with a triangle (orthocenter, incenter, Simson line, nine-point circle, etc.), quite a number of famous theorems (Ceva, Menelaus, Desargues, Pascal, Simson, Steiner, and more), some use of transformation-theoretic methods (homotheties, inversions), the use of complex numbers in geometry, a glimpse at three-dimensional geometry, and others.

The format of each chapter is roughly the same. There are some basic definitions and theorems, and some worked-out problems (referred to as “Delta” problems). Trigonometry is freely used in both the statements of theorems and the solutions to many of these problems. Then each chapter ends with roughly a dozen exercises (the so-called “Epsilon” problems), that are culled from competitions, journals and other sources. Solutions to these are not provided, and there is to my knowledge no instructor’s solution manual. References to the source are given for many of the Delta and Epsilon problems. (One quibble: on several occasions, the reference was to a journal, but the specific volume was omitted.)

By and large, the book is quite well written. I did notice an occasional mild glitch: the word “homothety”, for example, is not defined until page 199, but is used, several times, earlier. There were also things that I would have liked to have seen in this book that did not appear, chief among these being an Index. More and more, it seems, I am encountered mathematics books that lack an Index, and I think this is an appalling trend. It is particularly annoying in this book; a reader who wants to learn what the nine-point circle is, for example, will have a heck of a time finding a definition, seeing that the topic appears in a chapter whose title does not contain those words. (For the record, it is defined on page 100.)

On a mathematical level, I think it would have been nice to see some examples of isometries used to solve geometric problems, and to also see some problems concerning compass and straightedge constructions (though perhaps these topics don’t show up much in competitions).

Although this book does have significant overlap with the contents of a reasonable course in advanced course in Euclidean geometry, I think its primary value is not as a text: the problems, for example, are of Olympiad-caliber, and so probably are too difficult to assign as routine homework assignments in an average geometry course. I think the principal audience for this book would consist of people preparing for competitions, but students taking, or instructors teaching, a standard course in geometry, should also enjoy flipping through this book.

Mark Hunacek ( teaches mathematics at Iowa State University. He is giving serious thought to not adopting any book as a text if it lacks an Index.