Geometric Inequalities

This book would be more descriptively titled “How to Prove Geometric Inequalities”, as its focus is on proof techniques rather than on describing the most important inequalities. It is aimed at high-school students who want to participate in the Chinese and International Mathematical Olympiad (and their teachers), so it is not for everyone.

The book is largely a collection of worked examples. There are also lots of exercises, most having answers or hints in the back of the book. The book features a number of solutions by students, and appears in a series of problem books for the IMO.

Although the results covered here are indeed geometric, for the most part they assert inequalities between expressions calculated from measurements. A typical result that appears here (although an equality and not an inequality) is Ptolemy’s theorem about the sides and diagonals of a quadrilateral. The proofs depend on heavy use of algebra and trigonometry rather than on pure geometry.

This is a translation of a Chinese-language book that appeared in 2004. The translation is generally easy to read, but often is not in idiomatic English and goes off the rails from time to time. For example (p. 64): “So far the methods we used are mostly of geometric and triangular,” and “The following typical inequality was given by M. S. Klamkin at his early times.” Or (p. 106) “The above three answers are all good with respective peculiarity. As for the one offered by Long Yun, which is concise and throws light on people. Therefore it is somewhat the most excellent.”

Another good introductory book is Kazarinoff’s Geometric Inequalities. This follows a more geometrical approach (after Steiner), in which there is relatively little calculation and most of the results refer to one measurement being larger than another. But it is a much more elementary book and doesn’t cover much beyond the isoperimetric theorem and the reflection principle (mirror trick) for proving theorems.

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