Planimetry. A manual for in-depth study of mathematics

Planimetry. A manual for in-depth study of mathematics / V.F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, S. A. Shestakov, I. I. Yudina. - M., 2005. - 488 p.
This manual provides a systematic presentation of an in-depth course in planimetry. Along with the basic geometric information included in the standard school curriculum in geometry, it contains a large amount of 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, authors do not formulate theorems and axioms in advance, but search for their formulations together with the reader. This approach is explained by the authors’ desire to give an idea of ​​how mathematics is constructed and how mathematicians work.

The book pays considerable attention to Lobachevsky geometry, curves of constant width, isoperimetric problems, and proves a number of remarkable theorems of planimetry.

The manual is aimed at students who have 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 mathematical clubs and electives, and serve as the main textbook in schools specializing in physics and mathematics.
TABLE OF CONTENTS
Preface............................................. 3
Chapter 1. Basic geometric information................... 6
§ 1. Points, straight lines, segments.................................... 6
§2. Measuring segments and angles.................................... 17
§3. Perpendicular and parallel lines................... 25
Chapter 2. Triangles................................... 37
§ 1. Triangles and their types.................................... 37
§2. Isosceles triangle........................ 43
§3. Relationships between the sides and angles of a triangle....... 46
§4. Signs of equality of triangles.................................... 52
§5. Signs of equality of right triangles.......... 68
§6. Construction tasks......................................... 79
Chapter 3. Parallel lines.................................... 101
§1. Axiom of parallel lines........................ 101
§2. Properties of parallel lines................................... 119
Chapter 4. Further information about triangles.................... 127
§1. Sum of angles of a triangle. Middle line of the triangle...... 127
§2. Four Remarkable Points of the Triangle................................ 139
Chapter 5. Polygons................................... 150
§1. Convex polygon........................ 150
§2. Quadrilaterals................................... 168
Chapter 6. Area................................... 180
§1. Equal polygons.................................... 180
§2. The concept of area................................... 188
§3. Area of ​​a triangle................................... 197
§4. Heron's formula and its applications.................................... 210
§5. Pythagorean theorem................................... 213

Chapter 7. Similar triangles........................ 219
§1. Signs of similarity of triangles.................................... 219
§2. Application of similarity to theorem proving and problem solving. . 230
§3. Construction tasks......................................... 245
§4. About the remarkable points of the triangle................... 255
Chapter 8. Circle................................... 260
§1. Properties of a circle................................... 260
§2. Angles associated with a circle.................................... 268
Chapter 9. Vectors................................... 285
§1. Vector addition......................................... 285
§2. Multiplying a vector by a number........................ 292
Chapter 10. Coordinate method................................... 298
§ 1. Coordinates of points and vectors.................................... 298
§2. Equations of a line and a circle.................................... 304
§3. Radical axis and radical center of circles.......... 309
§4. Harmonic quadruples of points........................ 317
Chapter 11. Trigonometric relations in a triangle. Scalar product of vectors......................... 324
§ 1. Relationships between the sides and angles of a triangle....... 324
§2. Using trigonometric formulas in solving geometric problems.................................................331
§3. Dot product of vectors................................... 339
Chapter 12. Regular polygons. Length and area...... 347
§1. Regular polygons........................ 347
§2. Length........................................ 355
§3. Area......................................... 363
Chapter 13. Geometric transformations................................. 374
§1. Movements...................................... 374
§2. Central similarity................................... 386
§3. Inversion......................................... 396
Appendix 1. About numbers again*................................ 414
Appendix 2. Again about Lobachevsky’s geometry.................. 430

Butuzov Valentin Fedorovich

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 Fedorovich

head of department
Valentin Fedorovich Butuzov was born on November 23, 1939. in Moscow in a family of employees. Father, Butuzov Fedor Grigorievich (1909-1975) is a construction technician, mother, Butuzova (Kuraeva) Anastasia Vladimirovna (1912-1994) graduated from an art college and worked for many years as the head of a rural club. In 1957 V.F. Butuzov graduated with a gold medal from Sukharevskaya secondary school (Krasnopolyansky district, Moscow region) and entered the physics department of M.V. Lomonosov Moscow State University. Upon completion in 1963. was accepted into graduate school. The choice of specialty and the formation of scientific interests were greatly influenced by professors and teachers of the Department of Mathematics of the Faculty of Physics A.N. Tikhonov, A.G. Sveshnikov, A.B. Vasilyeva, P.S. Modenov. In 1966 graduated from graduate school, defended his PhD thesis “Asymptotics of solutions to some problems for integro-differential equations with a small parameter for derivatives” and was hired at the Department of Mathematics of the Faculty of Physics. Since 1970 annually gives general courses of lectures on higher mathematics, as well as a special course on asymptotic methods. In 1972 approved for the academic rank of associate professor. In 1979 defended his doctoral dissertation "Singularly perturbed boundary value problems with a corner boundary layer", in which he developed an effective method for constructing asymptotic expansions of solutions to a wide class of singularly perturbed problems in areas with corner boundary points.

Since 1981 works as a professor (the academic rank of professor was approved in 1982), since 1993. - Head of the Department of Mathematics, Faculty of Physics, Moscow State University.

Since 1979 V.F. 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 School Textbook Competition. Currently, tens of millions of schoolchildren in Russia and the CIS countries study using them. Under his editorship, two textbooks on higher mathematics for universities were written, which went through several editions and were translated into English and Spanish.

V.F. Butuzov was awarded the medals “For Labor Distinction” (1986) and “In Memory of the 850th Anniversary of Moscow” (1997), badges “Excellence in Public Education” (1985) and “Honored 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 trained 12 candidates of science, three of his students became doctors of science. In collaboration with Prof. A.B. Vasilyeva, he wrote four monographs on asymptotic methods in the theory of singular perturbations.

Main works:

1. Asymptotic expansions of solutions to singularly perturbed equations. M., Nauka, 1973 (together with A.B. Vasilyeva).
2. Asymptotic methods in the theory of singular perturbations. M., Higher School, 1990 (together with A.B. Vasilyeva).
3. Mathematical analysis in questions and problems. M., Higher School, 1st edition, 1984; M., Fizmatlit, 4th edition, 2001 (together with N.Ch. Krutitskaya, G.N. Medvedev, A.A. Shishkin).
4. Geometry 7-9 (textbook for general education institutions). M., Education, 1st edition, 1990; 15th edition, 2005 (together with L.S. Atanasyan, S.B. Kadomtsev, E.G. Poznyak, I.I. Yudina).
5. Geometry 10-11 (textbook for general education institutions). M., Education, 1st edition, 1992; 11th edition, 2005 (together with L.S. Atanasyan, S.B. Kadomtsev, L.S. Kiseleva, E.G. Poznyak).

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