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The present book is devoted to students of the last school grades, university students, teachers, lecturers 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. The significance of the complex numbers is far from a real recognition in known textbooks and scholar literature. The applications not only in mathematics but also in many other subjects are considerable and the present book is a strong proof of such a statement. 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 Mőbius transformation. The third chapter is dedicated to the geometry of circle and triangle on the base of complex numbers. Numerous theorems are proposed, namely: Menelau’s theorem, Pascal’s and Desargue’s theorem, Ceva’s and Van Aubel’s theorem, Stewart’s theorem, Ptolemy’s theorem and others. Exercises and problems are included in the Fourth chapter: 122 examples with solutions and 161 solved problems are 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.
A high just appraisal of the book is due to the numerous non-standard problems in it taken from the National Olympiads of Bulgaria, China, Iran, Japan, Korea, Poland, Romania, Russia, Serbia, Turkey, Ukraine and others but also from Several International and Balkan Mathematical Olympiads