Textbook in PDF and DJVU formats

Written as a supplement to Marcel Berger’s popular two-volume set, Geometry I and II (Universitext), this book offers a comprehensive range of exercises, problems, and full solutions. Each chapter corresponds directly to one in the relevant volume, from which it also provides a summary of key ideas. Where the original Geometry volumes tend toward challenging problems without hints, this book offers a wide range of material that begins at an accessible level, and includes suggestions for nearly every problem. Bountiful in illustrations and complete in its coverage of topics from affine and projective spaces, to spheres and conics, Problems in Geometry is a valuable addition to studies in geometry at many levels.
The textbook Geometry, published in French by CEDIC/Fernand Nathan and in English by Springer-Verlag (scheduled for 1985) was very favorably received. Nevertheless, many readers found the text too concise and the exercises at the end of each chapter too difficult, and regretted the absence of any hints for the solution of the exercises.
This book is intended to respond, at least in part, to these needs. The length of the textbook (which will be referred to as [B]throughout this book) and the volume of the material covered in it preclude any thought of publishing an expanded version, but we considered that it might prove both profitable and amusing to some of our readers to have detailed solutions to some of the exercises in the textbook.
At the same time, we planned this book to be independent, at least to a certain extent, from the textbook; thus, we have provided summaries of each of its twenty chapters, condensing in a few pages and under the same titles the most important notions and results used in the solution of the problems. The statement of the selected problems follows each summary, and they are numbered in order, with a reference to the corresponding place in [B]. These references are not meant as indications for the solutions of the problems. In the body of each summary there are frequent references to [B], and these can be helpful in elaborating a point which is discussed too cursorily in this book.
Following the summaries we included a number of suggestions and hints for the solution of the problems; they may well be an intermediate step between your personal solution and ours!
The bulk of the book is dedicated to a fairly detailed solution of each problem, with references to both this book and the textbook. Following the practice in [B], we have made liberal use of illustrations throughout the text.
Finally, I would like to express my heartfelt thanks to Springer-Verlag, for including this work in their Problem Books in Mathematics series, and to Silvio Levy, for his excellent and speedy translation.
Marcel Berger
Groups Operating on a Set: Nomenclature, Examples, Applications
Affine Spaces
Barycenters; the Universal Space
Projective Spaces
Affine-projective Relationship: Applications
Projective Lines, Cross-Ratios, Homographies
Complexifications
More about Euclidean Vector Spaces
Euclidean Affine Spaces
Triangles, Spheres, and Circles
Convex Sets
Polytopes; Compact Convex Sets
Quadratic Forms
Projective Quadrics
Affine Quadrics
Projective Conics
Euclidean Conics
The Sphere for Its Own Sake
Elliptic and Hyperbolic Geometry
The Space of Spheres