Boris Pritsker 
GEOMETRICAL KALEIDOSCOPE (2ND ED) 

The goal of the book is to provide insight into many enjoyable and fascinating aspects of geometry, and to reveal interesting geometrical properties. The emphasis is on the practical applications of theory in the problem-solving process. The chapters cover a myriad of topics among which are the classic theorems and formulas such as Archimedes’ Law of the Lever, the Pythagorean Theorem, Heron’s formula, Brahmagupta’s formula, Appollonius’s Theorem, Euler’s line properties, the Nine-Point Circle, Fagnano’s Problem, the Steiner-Lehmus Theorem, Napoleon’s Theorem, Ceva’s Theorem, Menelaus’s Theorem, Pompeiu’s Theorem, and Morley’s Miracle. The book focuses on geometric thinking — what it means, how to develop it, and how to recognize it. ‘Geometrical Kaleidoscope’ consists of a kaleidoscope of topics that seem to not be related at first glance. However, that perception disappears as you go from chapter to chapter and explore the multitude of surprising relationships, unexpected connections, and links. Readers solving a chain of problems will learn from them general techniques, rather than isolated instances of the application of a technique. In spite of the many problems’ challenging character, their solutions require no more than a basic knowledge covered in a high school geometry curriculum. There are plenty of problems for readers to work out for themselves (solutions are provided at the end of the book).

In the 2nd edition of the book there are many new ideas and additional explanations that help the reader better understand the solutions of problems and connect the chapters to one another. A new chapter ‘Alternative proofs of the Pythagorean Theorem’ is added. It covers seven different proofs of the famous theorem and discusses its generalizations and applications. There is also Appendix and Index added, which were missing in the first edition of the book.

Contents:

• Preface


• About the Author


• Medians, Centroid, and Center of Gravity of a System of Points


• Altitudes and the Orthocenter of a Triangle and Some of Its Properties


• The Orthic Triangle and Its Properties


• The Angle Bisector of a Triangle and Its Properties


• The Area of a Quadrilateral


• The Theorem of Ratios of the Areas of Similar Polygons


• A Pivotal Approach: Applying Rotation in Problem Solving


• Auxiliary Elements in Problem Solving


• Constructions Siblings


• Session of One Interesting Construction Problem


• Alternative Proofs of the Pythagorean Theorem


• Morley’s Theorem


• Solutions to the Problems and Exercises


• Appendix


• References


• Index

Readership: High school and college students, general public.

Key Features:

• The book offers a set of non-routine topics and problems in plane geometry that stimulate students to explore unfamiliar or little-known aspects of mathematics


• It also covers several classic historical problems and theorems and offered methods and techniques how to link them together


• The reader also learns how the exploring of different approaches for solving of the same problem might lead to discovering interesting generalizations and new unexpected properties and results


• There are many exercises for readers to solve (all with solutions and answers provided at the end of the book)


• The book shows how to make Geometry fun