Geometry is one of the most important and active fields in Mathematics with a substantial and large variety of applications in several disciplines, and with a very high impact in all levels of mathematical education.
This book deals with the essential results in plane Euclidean Geometry that are useful in solving difficult olympiad problems. The reader will become acquainted with well - known theorems such as Menelaus theorem, Ceva theorem, Ptolemy theorem, Stewart theorem, Euler nine point circle and the Euler line, etc., in the context of some complex geometric problems.
This book provides a very synthetic presentation of concepts and ideas in Euclidean Geometry, most of them without proof since its main goal is to illustrate by nonstandard problems how these ideas can be used. The book clearly demonstrates how instrumental it is to use various tools for the formulation of basic geometrical questions in order to find the simplest and the most intuitive arguments to solve a variety of problems. The book under review fully fits this purpose. In several situations and from different points of view the book presents the power of some natural geometric ideas. Most of the material is really suitable for advanced high-school classes and the book itself could offer a great service of attracting bright students to Mathematics.
The textbook is organized into six chapters. The first four chapters present some theoretical results including suggestive examples on the following aspects : Euclid's Elements, logic, methods of proof, fundamentals on geometric transformations and some important theoretical results in solving problems. Chapter 5 contains carefully selected Olympiad - caliber problems and it is organized into three sections : geometric problems with basic theory, geometric problems with more advanced theory, geometric inequalities.
The book concludes with a useful and relevant bibliography containing 99 references. It also contains an index of symbols and a subject index.
I would like to conclude this review with the statement of appreciation of the Fields Medalist Michael H. Freedman who wrote the foreword of the book :
`"........Young people need such texts, grounded in our shared intellectual history and challenging them to excel and create a continuity with the past. Geometry has seemed destined to give way in our modern computerized world to algebra. As with Michael Th. Rassias' previous homonymous book on number theory, it is a pleasure to see the mental discipline of the ancient Greeks so well represented to a youthful audience ".
All in all the text is a highly recommendable choice for any olympiad training program, and fills some gaps in the existing literature in Euclidean Geometry. The book is a very useful source of models and ideas for students, teachers, heads of national teams and authors of problems, as well as for people who are interested in mathematics and solving difficult problems.