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Methods for Euclidean Geometry: Second Edition nd Edition [Minkštas viršelis]

  • Formatas: Paperback / softback, 576 pages, aukštis x plotis x storis: 230x150x30 mm, weight: 800 g
  • Išleidimo metai: 26-Jun-2021
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486847268
  • ISBN-13: 9780486847269
Kitos knygos pagal šią temą:
  • Formatas: Paperback / softback, 576 pages, aukštis x plotis x storis: 230x150x30 mm, weight: 800 g
  • Išleidimo metai: 26-Jun-2021
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486847268
  • ISBN-13: 9780486847269
Kitos knygos pagal šią temą:
In this revised edition, the book presents classical solution methods for plane geometry problems, but its distinguishing feature is the subsequent wealth of methods it uses to solve problems, many of which arose where existing techniques proved inadequate.


Methods for Euclidean Geometry explores one of the oldest and most beautiful of mathematical subjects. In this revised edition, the book begins with a thorough presentation of classical solution methods for plane geometry problems, but its distinguishing feature is the subsequent wealth of methods it uses to solve problems, many of which arose where existing techniques proved inadequate. Students studying and applying this variety of methods, as well as the classic analytical method, will develop their appreciation of the subject and of mathematics as a whole.
Preface xi
1 Early History
1(9)
1.1 The Egyptians
2(1)
1.2 The Babylonians
3(2)
1.3 The Greeks
5(3)
1.4 Problems
8(2)
2 Axioms: From Euclid to Today
10(20)
2.1 Axiomatic Systems
10(4)
2.1.1 Components of an axiomatic system
11(1)
2.1.2 Models
12(2)
2.2 Euclidean and Non-Euclidean Geometries
14(5)
2.2.1 Euclid's axiomatic system
14(1)
2.2.2 Non-Euclidean geometries
15(2)
2.2.3 More on Euclid's "Elements"
17(2)
2.3 Alternate Axiom Sets
19(1)
2.3.1 Hilbert
19(1)
2.3.2 Birkhoff
20(1)
2.4 The SMSG Postulates
20(6)
2.5 Problems
26(1)
2.6
Chapter 2 Appendix
27(3)
3 Lines and Polygons
30(41)
3.1 Introduction
30(8)
3.1.1 Segments, angles, and polygons
30(3)
3.1.2 Similarity and congruence
33(2)
3.1.3 Special angles and parallel lines
35(3)
3.2 Triangles
38(24)
3.2.1 Angle relations in polygons
39(3)
3.2.2 Congruence of triangles
42(1)
3.2.3 Median, bisector, and altitude
43(1)
3.2.4 Isosceles triangles
44(2)
3.2.5 Basic triangle inequalities
46(1)
3.2.6 Similarity of triangles
47(7)
3.2.7 Homotheties
54(1)
3.2.8 Right triangles
54(4)
3.2.9 Problems
58(3)
3.2.10 Supplemental Problems
61(1)
3.3 Parallelograms and Trapezoids
62(9)
3.3.1 Parallelograms
62(3)
3.3.2 Trapezoids
65(2)
3.3.3 Problems
67(1)
3.3.4 Supplemental Problems
68(3)
4 Circles
71(33)
4.1 Definitions and Properties
71(12)
4.2 Inscribed Angles
83(8)
4.3 Lengths of Tangents and Chords
91(1)
4.4 Incircle and Circumcircle
92(6)
4.5 Problems
98(3)
4.6 Supplemental Problems
101(3)
5 Length and Area
104(27)
5.1 Area of a Polygonal Region
105(7)
5.2 Area and Circumference of a Circle
112(6)
5.2.1 A first approach to area
113(1)
5.2.2 A second approach to area
113(2)
5.2.3 Circumference of a circle
115(3)
5.3 Wallace-Bolyai-Gerwien Theorem
118(6)
5.4 Problems
124(5)
5.5 Supplemental Problems
129(2)
6 Loci
131(30)
6.1 Locus of a Property
131(4)
6.2 Three Notable Loci
135(16)
6.2.1 Ellipse
135(4)
6.2.2 Hyperbola
139(5)
6.2.3 Parabola
144(3)
6.2.4 History and comments
147(4)
6.3 Reflection Properties and Applications
151(4)
6.4 Problems
155(3)
6.5 Supplemental Problems
158(3)
7 Trigonometry
161(29)
7.1 A Short History
161(3)
7.2 Definitions and Properties
164(7)
7.3 Unit Circle Trigonometry
171(2)
7.4 Summary of Identities
173(5)
7.5 Applications
178(6)
7.6 Problems
184(3)
7.7 Supplemental Problems
187(3)
8 Coordinatization
190(30)
8.1 The Real Number Line
191(1)
8.2 Coordinates
192(4)
8.3 Equations of a Line
196(5)
8.4 Applications
201(6)
8.5 Systems of Equations, Revisited
207(4)
8.6 Equations of Circles
211(5)
8.7 Problems
216(2)
8.8 Supplemental Problems
218(2)
9 Conies
220(32)
9.1 Parabolas
221(4)
9.2 Ellipses
225(6)
9.3 Hyperbolas
231(5)
9.4 A Unified Definition of Conies as Loci
236(1)
9.5 Change of Coordinates
237(5)
9.6 The Graph of Ax2 + Bxy + Cy2 + Fx + Gy + H = 0
242(5)
9.7 Problems
247(2)
9.8 Supplemental Problems
249(3)
10 Complex Numbers
252(24)
10.1 Introduction
253(3)
10.2 Geometric Interpretation
256(7)
10.3 Applications
263(8)
10.4 Problems
271(2)
10.5 Supplemental Problems
273(3)
11 Vectors
276(27)
11.1 Introduction
276(1)
11.2 Computations
277(7)
11.3 Applications to Geometry
284(5)
11.4 The Dot Product
289(7)
11.5 Applications of the Dot Product
296(3)
11.6 Problems
299(2)
11.7 Supplemental Problems
301(2)
12 Affine Transformations
303(27)
12.1 Introduction
303(3)
12.2 Matrices
306(7)
12.3 Properties
313(6)
12.4 Applications
319(3)
12.5 Affine Transformations of Conic Sections
322(3)
12.6 Problems
325(2)
12.7 Supplemental Problems
327(3)
13 Inversions
330(24)
13.1 Introduction
331(1)
13.2 Basic Properties
332(4)
13.3 Theorems
336(4)
13.4 Classical Applications
340(9)
13.5 Problems
349(3)
13.6 Supplemental Problems
352(2)
14 Coordinate Method with Software
354(191)
Epilogue
359(2)
Hints to
Chapter Problems
361(1)
Triangles
361(2)
Parallelograms and Trapezoids
363(1)
Circles
364(2)
Length and Area
366(3)
Loci
369(1)
Trigonometry
370(1)
Coordinatization
371(1)
Conies
372(1)
Complex Numbers
373(1)
Vectors
374(1)
Affine Transformations
375(1)
Inversions
376(2)
Solutions to
Chapter Problems
378(1)
Early History
378(1)
Axioms: From Euclid to Today
379(6)
Triangles
385(16)
Parallelograms and Trapezoids
401(9)
Circles
410(21)
Length and Area
431(20)
Loci
451(14)
Trigonometry
465(16)
Coordinatization
481(11)
Conies
492(14)
Complex Numbers
506(11)
Vectors
517(9)
Affine Transformations
526(8)
Inversions
534(11)
Bibliography 545(4)
Index 549(6)
About the Authors 555