Elementary Geometry

This book is a translation of the German original “Elementargeometrie” published by F. Vieweg and Sohn Verlag in 2005. It is intended as a compendium of the curriculum of elementary geometry for university students. Chapters 1-2 are devoted to an exhaustive study of planar elementary geometry. Many classical theorems are demonstrated including the Ceva theorem, the Menelaus theorem and various properties of triangles, circles and conic sections. Chapter 3 describes the group of Euclidean transformations and its subgroups including discrete ones. Chapter 4 deals with hyperbolic geometry (mainly in the Poincaré model) and in chapter 5 basics of spherical geometry are developed. The main techniques of the book are based on analytical computations in the Euclidean plane, which is introduced as the two dimensional real affine plane with a positive definite quadratic form. Also its identification with the complex line is exploited. Unfortunately, we do not find in the book projective proofs of Euclidean results that are projective in their essence. The book is nicely written, with numerous figures, and the material in the book is organised systematically. It can be widely used by university students and teachers.

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