a guide for advanced study of mathematics online. Butuzov vf, kadomtsev s. b. planimetry. Online Math Advanced Study Guide The Inside of the Book
Moscow: Fizmatlit, 2005 .-- 488s.
This manual provides a systematic presentation of an advanced course in planimetry. Along with the basic geometric information included in the standard school curriculum on geometry, contains a large additional material, expanding and deepening the basic information. The style of presentation adopted in the manual differs markedly from the traditional one: theorem - proof. In a number of cases, the authors do not formulate theorems and axioms in advance, but seek their formulations together with the reader. This approach is explained by the desire of the authors to give an idea of \u200b\u200bhow mathematics is structured and how mathematicians work.
In the book, considerable attention is paid to Lobachevsky's geometry, curves of constant width, isoperimetric problems, a number of remarkable planimetry theorems are proved.
The manual is aimed at students with an increased interest in mathematics, as well as anyone who is attracted by the beauty of geometry. It can be used in classes with in-depth study of mathematics, in the work of math circles and electives, and serve as the main textbook in schools of physics and mathematics.
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Foreword 3
Chapter 1. Basic Geometric Information 6
§ 1. Points, lines, line segments 6
1. Point ( 6). 2. Straight line (b). 3. Ray and segment (9). 4. Multiple tasks A0). 5. Angle A3). b. Half-plane A4).
§2. Line and Angle Measurement 17
7. Equality geometric shapes A7). 8. Comparison of line segments and angles A7). 9. Midpoint and bisector of angle A8). 10. Measuring line segments and angles A9). 11. On the numbers B0).
§3. Perpendicular and parallel lines 25
12. Perpendicular lines B5). 13. Signs of parallelism of two straight lines B8). 14. Practical ways of constructing parallel lines C1). 15. Is there a square? C2). 16. Concluding remarks C4).
Chapter 2. Triangles 37
§ 1. Triangles and their types 37
17. Triangle C7). 18. Outside corner of triangle C8).
19. Classification of triangles C9). 20. Medians, bisectors and heights of the triangle D0).
§2. Isosceles triangle 43
21. The theorem on the angles of an isosceles triangle D3).
22. A sign of an isosceles triangle D3). 23. Theorem about the height of an isosceles triangle D4).
§3. Relations between the sides and angles of a triangle 46
24. The theorem on relations between the sides and angles of the triangle D6). 25. Converse Theorems D7). 26. The triangle inequality D9).
§4. Signs of equality of triangles 52
27. Three signs of equality of triangles E2). 28. Are there other signs of equality of triangles? E6). 29. Signs of equality of triangles using medians, bisectors and heights F1).
§5. Equality tests for right-angled triangles 68
30. Five signs of equality of right-angled triangles F8).
31. Mid perpendicular to the segment. Axial symmetry G2).
32. Distance from point to straight line G5). 33. Property of the angle bisector G5). 34. Theorem on the intersection of the bisectors of the triangle G7).
§6. Building tasks 79
35. Circle. Central symmetry G9). 36. Mutual arrangement of a straight line and a circle (81). 37. A circle inscribed in a triangle (84). 38. Mutual arrangement of two circles (85). 39. Construction of a triangle on three sides (88).
40. The main tasks for the construction (91). 41. A few more problems on the construction of a triangle (94).
Chapter 3. Parallel Lines 101
§ 1. Axiom of parallel lines 101
42. Axioms A01). 43. Basic concepts A02). 44. System of axioms for planimetry 45. Two consequences from axioms A08).
46. \u200b\u200bOn Theorems A09). 48. Axiom of parallel lines A14).
49. About the fifth postulate of Euclid A16). 50. Once again about the existence of the square A17).
§2. Properties of Parallel Lines 119
51. Distance between parallel lines A19). 52. Another way to construct parallel lines A20). 53. Tasks to build A21).
Chapter 4. Further Information on Triangles 127
§one. The sum of the angles of a triangle. Midline of triangle 127
54. The problem of cutting a triangle A27). 55. The sum of the angles of the triangle A29). 56. Midline of triangle A34). 57. Thales' theorem A34). 58. Surprising fact A36).
§2. Four Wonderful Points of a Triangle 139
59. The theorem on the intersection of the perpendicular to the sides of the triangle A39). 60. Circle circumscribed about triangle A41). 61. Theorem on the intersection of the heights of the triangle A42). 62. Reflections on the point of intersection of the medians of the triangle A43). 63. Theorem on the intersection of the medians of a triangle A45).
Chapter 5. Polygons 150
§ 1. Convex polygon 150
64. Broken line A50). 65. Polygon A52). 66. Convex polygon A58). 67. Convex line A61). 68. Closed line A62). 69. Closed convex line A63). 70. Inscribed polygon A64). 71. Described polygon A66).
§2. Quadrangles 168
72. Property of the diagonals of a convex quadrilateral (A68).
73. Characteristic property of the figure A70). 74. Parallelogram A70). 75. Theorems of Varignon and Gauss A72). 76. Rectangle, rhombus and square A73). 77. Trapezium A76).
Chapter 6. Area 180
§ 1. Equidistant polygons 180
78. Problems on cutting polygons A80). 79. composed polygons A83). 80. Cutting a square into unequal squares A85).
§2. The concept of area 188
81. Measuring the area of \u200b\u200ba polygon A88). 82. Area of \u200b\u200ban arbitrary figure A93).
§3. Area of \u200b\u200ba triangle 197
84. Areas of a rectangle, parallelogram and triangle A97). 85. Equal area polygons A98). 86. Euclidean B00 method). 87. Two theorems on the ratio of the areas of triangles B01). 88. Two theorems on the bisectors of the triangle B03). 89. The sign of equality of triangles on two sides and a bisector drawn from one vertex B04).
§4. Heron's formula and its applications 210
90. Heron's formula B10). 91. The median theorem B11). 92. Formula of the bisector of a triangle B12).
§5. Pythagoras' theorem 213
93. Generalized Pythagorean theorem B13). 94. The problem of cutting squares B15).
Chapter 7. Similar Triangles 219
§ 1. Tests for the similarity of triangles 219
95. Similarity and equality of triangles B19). 96. Other signs of similarity of triangles B22). 97. Trigonometric functions B24).
§2. Applying similarity to theorem proving and problem solving. ... 230
98. Thales' generalized theorem B30). 99. Corollary from the generalized Thales theorem B32). 100. Theorem about proportional segments in triangle B35). 101. Cheva's theorem B37).
102. Menelaus' theorem B41).
§3. Building Tasks 245
103. Geometric mean B45). 104. Arithmetic mean, harmonic mean and root mean square for two segments B46). 105. Similarity method B47).
§4. The Wonderful Points of a Triangle 255
106. On the heights of the triangle B55). 107. On the bisectors of the triangle B57). 108. Two more points associated with triangle B58).
Chapter 8. Circle 260
§ 1. Properties of the circle 260
109. Characteristic property of the circle B60). BY. Tasks for building B60). 111. Curves of constant width B63).
§2. Angles associated with a circle 268
112. Inscribed corners B68). 113. Angles between chords and secants B71). 114. Angle between tangent and chord B72). 115. The theorem on the square of the tangent B73). 116. Pascal's theorem B75).
117. Excircles of triangle B76).
Chapter 9. Vectors 285
§ 1. Addition of vectors 285
118. Co-directional vectors B85). 119. Equality of vectors B88). 120. Sum of vectors B89).
§2. Multiplying a vector by 292
121. Product of a vector by the number B92). 122. Multiple tasks B94).
Chapter 10. Coordinate Method 298
§ 1. Coordinates of points and vectors 298
123. Coordinate axis B98). 124. Rectangular coordinate system B99). 125. Coordinates of the vector C00). 126. Vector length and distance between two points C02). 127. Stewart's theorem C02).
§2. Equations of a Line and a Circle 304
128. Perpendicular vectors C04). 129. Equation of straight line C05). 130. Equation of the circle C06).
§3. Radical Axis and Radical Center of Circles 309
131. Radical axis of two circles C09). 132. Location of the radical axis relative to the circles C11). 133. Radical center of three circles C13). 134. Brianchon's theorem C15).
§4. Harmonic fours of points 317
135. Examples of harmonic fours C17). 136. Polar C20).
137. Quadruple C21). 138. Constructing a tangent line using one ruler C22).
Chapter 11. Trigonometric relations in a triangle. Dot product of vectors 324
§one. Relationship between sides and angles of a triangle 324
139. Sine and cosine of double angle C24). 140. Trigonometric functions of arbitrary angles C25). 141. Reduction formulas C25). 142. Another formula for the area of \u200b\u200ba triangle C26).
143. Sine theorem C27). 144. Cosine theorem C28).
§2. Using trigonometric formulas in solving geometric problems 331
145. Sine and cosine of the sum and difference of angles C31). 146. Morley's theorem C33). 147. Area of \u200b\u200ba quadrangle C35). 148. Areas of inscribed and circumscribed quadrangles C37).
§3. Dot product of vectors 339
149. Angle between vectors C39). 150. Definition and properties dot product vectors C41). 151. Euler's theorem C43). 152. Leibniz's theorem C44).
Chapter 12. Regular polygons. Length and area 347
§ 1. Regular polygons 347
153. Equilateral and conformal polygons C47).
154. Construction of regular polygons C50).
§2. Length 355
155. Circumference C55). 156. Length of line C57).
§ 3. Area 363
158. Area of \u200b\u200bfigure C63). 159. The first remarkable limit is C65). 160. Isoperimetric problem C67).
Chapter 13. Geometric Transforms 374
§ 1. Movements 374
161. Axial symmetry C74). 162. Movement C75). 163. Using movements in solving problems C77).
§2. Central likeness 386
164. Properties of central similarity C86). 165. Napoleon's theorem C88). 166. Euler's problem C89). 167. Simeon's line C92).
§3. Inversion 396
168. Definition of inversion C96). 169. Basic properties of inversion C98). 170. Ptolemy's theorem D01). 171. Euler's formula D02). 172. Circles of Apollonius D02). 173. Circles of Apollonius are needed even by filibusters (D05). 174. Feuerbach's theorem D07). 175. Problem of Apollonius D08).
Appendix 1. Again about numbers * 414
176. Non-negative real numbers D14). 177. Comparison of non-negative real numbers D17). 178. Addition of non-negative real numbers (D17). 179. Multiplication of positive real numbers (D18). 180. Negative real numbers D19). 181. Exact top edge D20).
182. Weierstrass theorem D21). 183. Binary notation of the number D21). 184. Oh mutual arrangement line and circle D23). 185. About measuring angles D26). 186. On the relative position of two circles D27).
Appendix 2. Again about the geometry of Lobachevsky 430
Answers and directions 437
Our notebook 471
Author Index 473
Index 474
From the Foreword:
This manual is aimed at students with an increased interest in mathematics, and is intended primarily for classes with advanced study of mathematics, for math circles and electives. It consists of 13 chapters corresponding to the chapters of the textbook "Geometry 7-9" by L.S. Atanasyan, V.F. Butuzov, SB. Kadomtseva, E.G. Poznyak, I.I. Yudina (Moscow: Education, 1990 and subsequent editions). At the same time, the manual is completely autonomous, which makes it possible to use it both in those classes where geometry is taught according to other textbooks, and as the main textbook in schools of physics and mathematics. It should be noted that the style of presentation adopted in the manual differs from the traditional one: a theorem is a proof. In a number of cases we do not formulate theorems and axioms in advance, but seek their formulations together with the reader. This approach is explained by the desire of the authors to give an idea of \u200b\u200bhow mathematics is structured and how mathematicians work.
The manual, along with the basic geometric information included in the standard school curriculum for geometry, contains a large additional material that expands and deepens the basic information. In particular, considerable attention is paid to the theory of parallel lines and an idea of \u200b\u200bthe geometry of Lobachevsky related to it is given.
In each chapter, as the theoretical material is presented, problems with solutions are given, illustrating the application of certain statements. For each paragraph of the chapter, tasks are given for independent workprovided with answers and directions. The most difficult tasks and sections are marked with an asterisk. There is also a subject index that makes it easy to navigate the book. We hope that our book will be of interest not only for teachers and students in advanced mathematics classes, but also for everyone who is attracted by the beauty of geometry.
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The manual is aimed at students with an increased interest in mathematics, as well as anyone who is attracted by the beauty of geometry. It can be used in advanced math classes, in work ...
This manual provides a systematic presentation of an advanced planimetry course. Along with the basic geometric information included in the standard school curriculum in geometry, there is a large additional material that expands and deepens the basic information. The style of presentation adopted in the manual differs markedly from the traditional one: theorem - proof. In a number of cases, the authors do not formulate theorems and axioms in advance, but seek their formulations together with the reader. This approach is explained by the desire of the authors to give an idea of \u200b\u200bhow mathematics is structured and how mathematicians work.
In the book, considerable attention is paid to Lobachevsky's geometry, curves of constant width, isoperimetric problems, a number of remarkable planimetry theorems are proved.
The manual is aimed at students with an increased interest in mathematics, as well as anyone who is attracted by the beauty of geometry. It can be used in classes with in-depth study of mathematics, in the work of math circles and electives, and serve as the main textbook in schools of physics and mathematics.
2nd edition, stereotyped.
Butuzov Valentin Fyodorovich
The department employs 55 teachers and researchers, including 13 professors and 19 associate professors, 17 employees of the department are doctors and 36 are candidates of science.
Butuzov Valentin Fyodorovich
head of departmentValentin Fedorovich Butuzov was born on November 23, 1939. in Moscow in a family of employees. Father, Butuzov Fedor Grigorievich (1909-1975), a construction technician, mother, Butuzova (Kuraeva) Anastasia Vladimirovna (1912-1994) graduated from an art college and for many years worked as the head of a village club. In 1957. V.F.Butuzov graduated with a gold medal from the Sukharev secondary school (Krasnopolyansky district of the Moscow region) and entered the physics faculty of Lomonosov Moscow State University. After graduating in 1963. was admitted to graduate school. Professors and teachers of the Department of Mathematics of the Faculty of Physics A.N. Tikhonov, A.G. Sveshnikov, A.B. Vasilieva, P.S.Modenov greatly influenced the choice of specialty and the formation of scientific interests. In 1966. graduated from graduate school, defended his Ph.D. thesis "Asymptotics of solutions of some problems for integro-differential equations with a small parameter at the derivatives" and was employed at the Department of Mathematics of the Physics Faculty. Since 1970. annually reads general courses of lectures on higher mathematics, as well as a special course on asymptotic methods. In 1972. approved in the academic rank of associate professor. In 1979. defended his doctoral dissertation "Singularly perturbed boundary value problems with an angular boundary layer", in which an effective method was developed for constructing asymptotic expansions of solutions of a wide class of singularly perturbed problems in domains with corner points of the boundary.
Since 1981 works as a professor (the academic title of professor was approved in 1982), since 1993. - Head of the Department of Mathematics, Faculty of Physics, Moscow State University.
Since 1979 VF Butuzov, together with his colleagues, takes an active part in the creation of new school textbooks on geometry. In 1988. these textbooks (for grades 7-9 and grades 10-11) took 1st place in the All-Union competition of school textbooks. Currently, tens of millions of schoolchildren in Russia and the CIS countries study using them. Two were written under his editorship teaching aids in higher mathematics for universities, withstood several editions and translated into English and Spanish.
VF Butuzov was awarded medals "For Labor Distinction" (1986) and "In Commemoration of the 850th Anniversary of Moscow" (1997), badges "Excellence in Public Education" (1985) and "Honorary Worker of Higher Professional Education of the Russian Federation" (1999). He is a laureate of the Lomonosov Prize of Moscow State University for teaching activities (1993), laureate of the Lomonosov Prize of Moscow State University, 1st degree for scientific work (2003).
He prepared 12 candidates of sciences, three of his students became doctors of sciences. In collaboration with Prof. A.B. Vasilyeva, he has written four monographs on asymptotic methods in the theory of singular perturbations.
Major works:
- Asymptotic Expansions of Solutions to Singularly Perturbed Equations (Moscow, Nauka, 1973) (with A.B. Vasil'eva).
- Asymptotic Methods in the Theory of Singular Perturbations), Moscow, Vysshaya Shkola, 1990 (with A.B. Vasilyeva).
- Mathematical Analysis in Questions and Problems), Moscow, Higher School, 1st edition, 1984; Moscow, Fizmatlit, 4th edition, 2001 (together with N. Ch.Krutitskaya, G.N. Medvedev, A.A. Shishkin).
- Geometry 7-9 (textbook for educational institutions) .M., Education, 1st edition, 1990; 15th edition, 2005 (together with L.S. Atanasyan, S. B. Kadomtsev, E. G. Poznyak, I.I. Yudina).
- Geometry 10-11 (textbook for educational institutions) .M., Education, 1st edition, 1992; 11th edition, 2005 (together with L.S. Atanasyan, S.B. Kadomtsev, L.S. Kiseleva, E.G. Poznyak).
