This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.114633655Institut des Hautes Etudes Scientifiques (IHES)Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG)Max-Planck-GesellschaftFAPEMIG [00163/06]