The literal translation of the title of this book is "Revisited Geometry?". The authors, as it were, guide the reader to the most beautiful places of an ancient, but ageless country? Geometry. Some mathematicians have recently begun to classify geometry as a secondary mathematical field. Book G. C. M. Coxeter and C. L. Greitzer is a bright document in defense of geometry, for asserting geometry in its proper place in the school system. At the same time, it is an excellent material for the work of school mathematical circles .. The study of this book gives an opportunity to look at geometry as a whole and at the same time to get acquainted with its individual pearls. The book, although it contains many tasks, is written in the usual manner of sequential presentation of the material. At the same time, the authors have saturated the presentation with a large number of interesting information on the history of the appearance of ideas and results, which makes the book even more attractive. Content. From the editor of Russian translation . Foreword by . Chapter 1. POINTS AND LINES RELATED TO THE TRIANGLE . ? 1. Generalized Sinus Theorem. ? 2. Cheva's theorem . ? 3. Great points . ? 4. Inscribed and indented circles. ? 5. Steiner - Lemus theorem . ? 6. Orthotriangle. ? 7. Mid-Triangle and Euler Line. ? 8. Circle of nine points . ? 9. Pedal Triangle . Chapter 2. SOME ENVIRONMENTAL PROPERTIES . ? 1. Degree of point relative to circle. ? 2. Radical axis of two circles . ? 3. Coaxial circles. ? 4. Again on the heights and orthocenter of the triangle. ? 5. Straight Symsons. ? 6. Ptolemy's Theorem . ? 7. Once again about the straight simson . ? 8. The Butterfly Theorem . ? 9. Morley's Theorem . Chapter 3. COLLINERITY AND COMPETITIVENESS . ? 1. Quadrilaterals; Varignon's Theorem. ? 2. Inscribed quadrilaterals; Brahmagupta theorem. ? 3. Napoleon's Triangles . ? 4. Menelaus Theorem . ? 5. Papp's theorem. ? 6. Perspective triangles; Desargues' theorem. ? 7. Hexagons. ? 8. Pascal's Theorem. ? 9. Brianchon's Theorem . Chapter 4. TRANSFORMATION . ? 1. Parallel Transfer. ? 2. Turn. ? 3. Reversal. ? 4. Symmetry. ? 5. Phaniano's task. ? 6. The problem of three jugs . ? 7. Dilation. ? 8. Spiral Similarity. ? 9. Genealogy of change. Chapter 5. INTRODUCTION TO INVERSIVE GEOMETRY . ? 1. Breakdown. ? 2. Difficult relationship. ? 3. Inversion. ? 4. Circular plane. ? 5. Orthogonality. ? 6. Feuerbach's Theorem . ? 7. Coaxial circles. ? 8. Inverse distance. ? 9. Hyperbolic Functions . Chapter 6. INTRODUCTION TO PROJECTIVE GEOMETRY . ? 1. Polar transformation. ? 2. The Polar Circle of a Triangle . ? 3. Conic Sections. ? 4. Focus and Directress. ? 5. Projective plane. ? 6. Central conical sections. ? 7. Stereographic and Gmonomic Projections . Answers and instructions for exercises . Bibliography . Dictionary of basic terms used in the book. Index. Other issues of the series on the website. Issue. 1 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Arithmetic and Algebraic . Issue. 2 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Geometry (planimetry), Vyp. 3 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Geometry (stereometry), Vyp. 4 - Boltyansky V. G. , Yaglom I. M. Convex Figures . Issue. 5 - Yaglom I. M. , Yaglom A. M. Non-elementary tasks in elementary presentation. Issue. 6 - Dynkin E. B. , Uspensky V. A. Mathematical Conversations . Issue. 7 - Yaglom I. M. Geometric transformations. Volume 1. Issue. 8 - Yaglom I. M. Geometric transformations. Volume 2. Issue. 9 - Balk M. B. Geometric applications of the concept of center of gravity. Issue. 10 - Rademacher G. , Tplic O. Numbers and Figures . Issue. 11 - Yaglom I. M. Galileo's Principle of Relativity and Non-Euclidean Geometry . Issue. 12 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Geometric inequalities and maximum and minimum tasks. Issue. 13 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Geometric estimates and problems from combinatorial geometry