Mathematics via Problems: Part 2: Geometry
- Length: 177 pages
- Edition: 1
- Language: English
- Publisher: American Mathematical Society
- Publication Date: 2021-08-24
- ISBN-10: 1470448793
- ISBN-13: 9781470448790
- Sales Rank: #707103 (See Top 100 Books)
This book is a translation from Russian of Part II of the book Mathematics Through Problems: From Olympiads and Math Circles to Profession. Part I, Algebra, was recently published in the same series. Part III, Combinatorics, will be published soon. The main goal of this book is to develop important parts of mathematics through problems. The authors tried to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover and recreate much of elementary mathematics and start edging into more sophisticated topics such as projective and affine geometry, solid geometry, and so on, thus building a bridge between standard high school exercises and more intricate notions in geometry. Definitions and/or references for material that is not standard in the school curriculum are included. To help students that might be unfamiliar with new material, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions. The book is based on classes taught by the authors at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Cover Title page Foreword Problems, exercises, circles, and olympiads Why this book, and how to use it English-language references Introduction What this book is about and who it is for Learning by solving problems Parting words By A.Ya.Kanel-Belov Olympiads and mathematics Research problems for high school students How this book is organized Resources and literature Acknowledgments Numbering and notation Notation References Chapter 1. Triangle 1. Carnot’s principle (1) By V.Yu.Protasov and A.A.Gavrilyuk Suggestions, solutions, and answers 2. The center of the inscribed circle (2) By V.Yu.Protasov Suggestions, solutions, and answers 3. The Euler line By V.Yu.Protasov Suggestions, solutions, and answers 4. Carnot’s formula (2*) By A.D.Blinkov Suggestions, solutions, and answers 5. The orthocenter, orthotriangle, and nine-point circle (2) By V.Yu.Protasov Suggestions, solutions, and answers 6. Inequalities involving triangles (3*) By V.Yu.Protasov Suggestions, solutions, and answers 7. Bisectors, heights, and circumcircles (2) By P.A.Kozhevnikov Suggestions, solutions, and answers 8. \enquote{Semi-inscribed} circle (3*) By P.A.Kozhevnikov Main series of problems—1 Main series of problems—2 Supplementary problems—1 Supplementary problems—2 Suggestions, solutions, and answers 9. The generalized Napoleon’s theorem (2*) By P.A.Kozhevnikov Introductory problems Formulation and proof of the generalized Napoleon’s theorem Suggestions, solutions, and answers 10. Isogonal conjugation and the Simson line (3*) By A.V.Akopyan Suggestions, solutions, and answers Additional reading Chapter 2. Circle 1. The simplest properties of a circle (1) By A.D.Blinkov Suggestions, solutions, and answers 2. Inscribed angles (1) By A.D.Blinkov and D.A.Permyakov Suggestions, solutions, and answers 3. Inscribed and circumscribed circles (2) By A.A.Gavrilyuk Suggestions, solutions, and answers 4. The radical axis (2) By I.N.Shnurnikov and A.I.Zasorin 5. Tangency (2) By I.N.Shnurnikov and A.I.Zasorin 6. Ptolemy’s and Casey’s Theorems (3*) By A.D.Blinkov and A.A.Zaslavsky 6.A. Ptolemy’s Theorem 6.B. Casey’s Theorem Suggestions, solutions, and answers Chapter 3. Geometric transformations 1. Applications of transformations (1) By A.D.Blinkov Suggestions, solutions, and answers 2. Classification of isometries of the plane (2) By A.B.Skopenkov Hints 3. Classification of isometries of space (3*) By A.B.Skopenkov Hints 4. An application of similarity and homothety (1) By A.D.Blinkov Suggestions, solutions, and answers 5. Rotational homothety (2) By P.A.Kozhevnikov 5.A. Introductory problems involving cyclists 5.B. Main problems 5.C. Additional problems Suggestions, solutions, and answers 6. Similarity (1) By A.B.Skopenkov 7. Dilation to a line (2) By A.Ya.Kanel-Belov Suggestions, solutions, and answers 8. Parallel projection and affine transformations (2) By A. B. Skopenkov Suggestions, solutions, and answers 9. Central projection and projective transformations (3) By A. B. Skopenkov 10. Inversion (2) By A. B. Skopenkov Additional reading Chapter 4. Affine and projective geometry 1. Mass points (2) By A.A.Gavrilyuk Suggestions, solutions, and answers 2. The cross-ratio (2) By A.A.Gavrilyuk Suggestions, solutions, and answers 3. Polarity (2) By A.A.Gavrilyuk and P.A.Kozhevnikov Fundamental properties and introductory problems Main problems Additional problems Suggestions, solutions, and answers Additional reading Chapter 5. Complex numbers and geometry (3) By A.A.Zaslavsky 1. Complex numbers and elementary geometry Suggestions, solutions, and answers 2. Complex numbers and Möbius transformations Additional problems Suggestions, solutions, and answers Additional reading Chapter 6. Constructions and loci 1. Loci (1) By A.D.Blinkov Suggestions, solutions, and answers 2. Construction and loci problems involving area (1) By A.D.Blinkov Suggestions, solutions, and answers 3. Construction toolbox (2) By A.A.Gavrilyuk Suggestions, solutions, and answers 4. Auxiliary constructions (2*) By I.I.Shnurnikov Suggestions, solutions, and answers Additional reading Chapter 7. Solid geometry 1. Drawing (2) By A.B.Skopenkov Suggestions, solutions, and answers 2. Projections (2) By M.A.Korchemkina 2.A. Projections of figures constructed from cubes 2.B. Trajectories 3. Regular polyhedra (3) 3.A. Inscribed and circumscribed polyhedra By A.Ya.Kanel-Belov Suggestions, solutions, and answers 3.B. Symmetries By A.B.Skopenkov 4. Higher-dimensional space (4*) By A.Ya.Kanel-Belov 4.A. Simplest polyhedra in higher-dimensional space By Yu.M.Burman and A.Ya.Kanel-Belov 4.B. Multi-dimensional volumes 4.C. Volumes and intersections 4.D. Research problems 4.E. Partitions into parts of smaller diameter By A.M.Raigorodsky Suggestions, solutions, and answers Additional reading Chapter 8. Miscellaneous geometry problems 1. Geometric optimization problems (2) By A.D.Blinkov Suggestions, solutions, and answers 2. Area (2) By A.D.Blinkov Suggestions, solutions, and answers 3. Conic sections (3*) By A.V.Akopyan Suggestions, solutions, and answers 4. Curvilinear triangles and non-Euclidean geometry (3*) By M.B.Skopenkov Additional problems Suggestions, solutions, and answers Additional reading Bibliography Index Back Cover
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