Geometry of Complex Numbers

The present book is devoted to students of the last school grades, university students, teachers, lectures and all lovers of mathematics who want to enrich their knowledge and skills in complex numbers and their numerous applications in Euclidean Geometry. Few countries in the world include complex numbers in their secondary school curriculum but even if included the volume of the corresponding content is quite insufficient consisting of elementary operations and geometric representation at most.

Mainly, the book will be useful for outstanding students with high potentialities in mathematics preparing themselves for successful participation in mathematical competitions and Olympiads. Other target groups are not excluded, namely those, whose representatives like to meet real challenges, connected with unexpected circumstances in problem solving.

The material in the book is divided into four chapters. The first one contains basic properties of the complex numbers, their algebraic notation, the notion of a conjugate complex number, geometric, trigonometric and exponential presentations, also interesting facts in connection with Reimann interpretation and the set Cⁿ. The second chapter includes various transformations of complex numbers in the Euclidean plane like similarity, homothety, inversion and Mоbius transformation. The third chapter is dedicated to the geometry of circle and triangle on the base of complex numbers. Exercises and problems are included in the Fourth chapter: 122 examples with solutions and 161 solved problems pare proposed. Together with all the 138 theorems, lemmas and corollaries accompanied by 64 examples and 88 figures the book turns out to be a rather exhaustive collection of the complex number applications in Euclidean Geometry.

Preface
Complex numbers
Transformation in Eucledian plane
Geometry of circle and triagle
Examples and exercises