Geometric Inequalities: Methods of Proving

This is a collection of about 1,000 problems in geometric inequalities, with complete solutions for about 900 of them. The problems are the sort that appear in the International Mathematical Olympiads, and the book is aimed primarily at students participating in the IMO.

The term “geometric inequality” is not defined, but appears to mean inequalities about geometric measurements. Most of the figures involved are triangles, although there is also a good bit on general polygons and on circles. For the most part, it deals with figures in the plane. A modest number of inequalities are formulas involving trigonometric functions, for which no illustration is given. The problems are organized according to the method used to solve them, so, for example, Chapter 1 is about problems solved using the triangle inequality. Chapter 5 claims to be about applying trigonometric inequalities, but in fact, is about applying trigonometric identities followed usually by quadratic inequalities. Generally speaking, the inequalities are not very interesting in themselves, but they present challenges for the student to overcome. One conspicuous omission (at least I could not find it) is the isoperimetric property.  (Very Bad Feature: no index.)

The prerequisites are not stated, but appear to be high-school math without calculus. It does assume familiarity with the Arithmetic Mean–Geometric Mean inequality, but not with the Cauchy–Schwarz (or dot-product) inequality, that would have been handy in problem 5.4.4. It assumes familiarity with Stewart’s Theorem about the cevian, which I was not familiar with.

The book seems to contain few factual errors, although there are some formatting glitches. The most noticeable is that the symbols for the trigonometric functions in Chapter 5 are typeset inconsistently, sometimes in roman (good) and sometimes in italic (bad), and in at least one case (5.1.2) a mixture of roman and italic in the same symbol. At least one of the citations of Stewart’s Theorem spells it Stuart (p. 109). The drawings on p. 432 are squashed slightly, so the circles are not circular.

The authors have another book, Algebraic Inequalities, that has the same general structure and some of the same strengths and weaknesses. The difference is that there the inequalities are based on algebraic manipulation or a little calculus rather than geometry or trigonometry.

I believe this book is well-suited for its intended purpose, which is teaching how to solve IMO-type problems. But I believe it doesn’t teach very much about the most useful sorts of inequalities. A good book for that, although very elementary, is Kazarinoff’s Geometric Inequalities. An all-around good book, though more advanced than the present one and not very geometrical, is Steele’s The Cauchy–Schwarz Master Class.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.