Mathematics via problems: Part 2 Geometry
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| Format: | Buch |
|---|---|
| Sprache: | English |
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American Mathematical Society
Providence, Rhode Island
[2021]
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| Schriftenreihe: | MSRI mathematical circles library
26 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xx, 177 Seiten Diagramme |
| ISBN: | 9781470448790 |
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Datensatz im Suchindex
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| adam_text | Contents Foreword Problems, exercises, circles, and olympiads Why this book, and how to use it English-language references ix ix x xi 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 xiii xiii xiv xv xv xvi xvi xvi xvii xvii xviii xix 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 11 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 1 2 3 4 5 7 7 8 9 13 13 14 15 16 V
vi CONTENTS 8. “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 19 19 20 20 20 21 25 25 26 27 31 33 38 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 39 39 40 42 44 46 47 47 49 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. В. 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, andanswers 5. Rotational homothety (2)ByP. A. Kozhevnikov 5. A. Introductory
problems involving cyclists 5. B. Main problems 5. C. Additional problems 55 55 57 61 63 63 65 50 50 51 52 65 67 71 71 72 73
CONTENTS vii 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 73 76 77 78 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 87 87 89 90 92 93 93 94 95 96 98 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 99 99 101 102 103 103 104 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 1.1. Shnurnikov Suggestions, solutions, and answers Additional reading 105 105 106 Chapter 7. Solid geometry 1. Drawing (2) By A. В. Skopenkov
125 125 78 80 81 83 86 111 113 117 119 120 121 123
viii CONTENTS Suggestions, solutions, and answers 2. Projections (2) By Μ. 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. В. Skopenkov 4. Higher-dimensional space (4*) By A. Ya. Kanel-Belov 4. A. Simplest polyhedra in higher-dimensional space By Yu. Μ. 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. Μ. Raigorodsky Suggestions, solutions, and answers Additional reading 126 126 126 128 130 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 145 145 147 151 152 158 161 Bibliography 171 Index 175 130 132 132 134 134 138 139 140 141 141 143 166 168 169 169
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| adam_txt |
Contents Foreword Problems, exercises, circles, and olympiads Why this book, and how to use it English-language references ix ix x xi 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 xiii xiii xiv xv xv xvi xvi xvi xvii xvii xviii xix 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 11 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 1 2 3 4 5 7 7 8 9 13 13 14 15 16 V
vi CONTENTS 8. “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 19 19 20 20 20 21 25 25 26 27 31 33 38 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 39 39 40 42 44 46 47 47 49 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. В. 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, andanswers 5. Rotational homothety (2)ByP. A. Kozhevnikov 5. A. Introductory
problems involving cyclists 5. B. Main problems 5. C. Additional problems 55 55 57 61 63 63 65 50 50 51 52 65 67 71 71 72 73
CONTENTS vii 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 73 76 77 78 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 87 87 89 90 92 93 93 94 95 96 98 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 99 99 101 102 103 103 104 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 1.1. Shnurnikov Suggestions, solutions, and answers Additional reading 105 105 106 Chapter 7. Solid geometry 1. Drawing (2) By A. В. Skopenkov
125 125 78 80 81 83 86 111 113 117 119 120 121 123
viii CONTENTS Suggestions, solutions, and answers 2. Projections (2) By Μ. 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. В. Skopenkov 4. Higher-dimensional space (4*) By A. Ya. Kanel-Belov 4. A. Simplest polyhedra in higher-dimensional space By Yu. Μ. 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. Μ. Raigorodsky Suggestions, solutions, and answers Additional reading 126 126 126 128 130 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 145 145 147 151 152 158 161 Bibliography 171 Index 175 130 132 132 134 134 138 139 140 141 141 143 166 168 169 169 |
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| spelling | Ėlementy matematiki v sadačach Mathematics via problems Part 2 Geometry Arkadiy Skopenkov American Mathematical Society Providence, Rhode Island [2021] xx, 177 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier MSRI mathematical circles library 26 MSRI Mathematical circles library Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Skopenkov, Arkadiy 1972- Sonstige (DE-588)1232352764 oth Zaslavskij, Aleksej A. 1960- Sonstige (DE-588)143267965 oth (DE-604)BV047830007 2 Erscheint auch als Online-Ausgabe 978-1-4704-6521-6 MSRI mathematical circles library 26 (DE-604)BV024626790 26 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033213361&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title_auth | Mathematics via problems |
| title_exact_search | Mathematics via problems |
| title_exact_search_txtP | Mathematics via problems |
| title_full | Mathematics via problems Part 2 Geometry Arkadiy Skopenkov |
| title_fullStr | Mathematics via problems Part 2 Geometry Arkadiy Skopenkov |
| title_full_unstemmed | Mathematics via problems Part 2 Geometry Arkadiy Skopenkov |
| title_short | Mathematics via problems |
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