This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. It starts with classical Euclidean geometry and its basic questions (axioms, basic constructions, the Thales and Pythagoras theorems). Coordinates and algebra bring new, useful tools and allow the formation of a general notion of a vector space with a given scalar product. Questions connected with perspective drawings lead to a discussion of projective geometry (with a more detailed discussion of the projective plane). The associated transformation group is presented as an example of Klein’s approach to geometry and his Erlangen program. The last chapter treats non-Euclidean geometries from the Klein point of view. The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers.

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