دانلود کتاب Geometry

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Contents......Page 5
for the series of books written by Israel Gelfand for high-school students......Page 8
What is special about this book? Why and for whom was it written?......Page 10
About the process of writing Geometry.......Page 11
Acknowledgements.......Page 14
Geometry is the simplest model of spatial relationships in our world......Page 15
Structure of this book and how to read it......Page 17
1.1 What is a point and what is a line?......Page 20
1.2 Operations available in Chapter I......Page 22
1.3 Ray, segment, half-plane......Page 23
1.4 Constructions with a straightedge......Page 25
2.1 Notion of an angle......Page 29
2.2 Some types of angles......Page 32
3.1 Configurations of three lines......Page 35
3.2 Triangles......Page 40
4 Four lines. Quadrilaterals......Page 44
6 Projection from a point onto a line......Page 48
7 Dual configurations in projective geometry......Page 59
8 Desargues configuration......Page 64
9 Dual Desargues configuration......Page 69
10 Algebraic notation or “computer presentation” of configurations......Page 75
11 Polygons and n straight lines......Page 78
12 Convex polygons, convex hull of n points......Page 81
13 Solution of Exercise 3 with the help of a Desargues configuration......Page 84
14 Overview of Chapter I......Page 89
1 Parallel straight lines......Page 90
2 Operations available in Chapter II......Page 92
3.1 Transitivity of parallel lines......Page 93
3.3 Reflexivity of parallel lines......Page 94
4.1 Equality of segments lying on parallel lines......Page 95
4.2 Construction of equal segments on parallel lines......Page 98
Properties of equal segments lying on parallel lines......Page 100
4.3 Construction of a segment of double length......Page 102
4.4 Division of a segment into equal parts......Page 103
5.1 Definition of a parallelogram......Page 106
5.2 Properties of parallelograms......Page 107
5.3 Proof of the Lemma......Page 112
5.4 More properties of parallelograms......Page 115
6.1 Bimedian of a triangle......Page 118
6.2 Median of a triangle......Page 124
7 Trapezoids......Page 126
8 The Minkowsky addition of two figures......Page 132
9 Parallel projection......Page 134
10.1 Parallel translation of a figure......Page 139
Sum of the exterior angles of a polygon......Page 145
Defining the same parallel translation by indicating different pairs of points......Page 147
10.3 Parallel translation on a line......Page 149
11 Central symmetry on the plane......Page 151
11.1 Sequences of parallel translations and central symmetries. The relation between central symmetry and parallel translation......Page 156
12.1 Vectors and parallel translations......Page 164
12.2 Addition of vectors......Page 165
12.3 Vectors lying on parallel lines......Page 170
12.4 Subtraction of vectors......Page 173
12.5 More problems on vectors......Page 175
13 Overview of Chapter II......Page 178
1 Why we cannot define equal segments in Chapter II......Page 180
2.1 Variation of the Desargues configuration in the case of parallel lines......Page 183
2.3 A property of parallel translation......Page 186
3.1 Addition and subtraction......Page 189
3.2 Multiplication and division......Page 192
4.1 Number axis......Page 196
4.2 Finding the coordinate of a point and length of a segment......Page 199
5 Affine coordinate systems on the plane......Page 200
1 The area of a figure......Page 208
2.1 Constructing parallelograms with rational area......Page 213
Changing the length of the sides of a unit parallelogram......Page 218
Changing the direction of the sides of a unit parallelogram......Page 220
2.3 How to measure the area of a parallelogram......Page 223
3 Area of a triangle......Page 224
4 Area of a trapezoid......Page 234
5 Area of a polygon......Page 236
6 More problems on areas......Page 243
7 How to measure the area of a figure......Page 246
8 Overview of Chapter III......Page 249
1 Operations available in Chapter IV......Page 250
1.1 Properties of a circle. Some related definitions......Page 252
2 Comparing segments......Page 254
3.1 Comparing angles. Degree measure......Page 256
Arc degree measure......Page 257
3.3 Addition of angles......Page 260
3.4 Vertical angles and angles with respectively parallel sides......Page 267
4.1 Turns and reflections......Page 271
4.2 Consecutive operations with a figure. Congruent figures......Page 273
5 Elements of a triangle. Congruent triangles......Page 276
6 Construction of a triangle from its elements......Page 278
Additional constructions of a triangle from its elements......Page 280
7.1 Relations between the sides of a triangle......Page 282
7.2 Relations between the angles of a triangle......Page 285
7.3 More about angles in a triangle......Page 287
8 Properties of a triangle. Particular kinds of triangles......Page 290
8.1 The isosceles triangle......Page 292
8.2 Equilateral triangle......Page 295
8.3 Right triangle......Page 296
9.1 Measurement of area. Area of a rectangle......Page 300
9.2 Area of a triangle......Page 301
10.1 The Pythagorean theorem......Page 304
10.2 The use of the Pythagorean theorem in arbitrary triangles......Page 309
10.3 Heron’s formula for the area of a triangle......Page 313
11.1 Perpendicular from a point to a line......Page 315
11.2 Distance from a point to a line......Page 317
11.3 The locus of points lying at equal distance from two given points......Page 318
11.4 The locus of points lying at equal distance from two given lines. Two definitions of an angle bisector......Page 320
11.5 Angles with respectively perpendicular sides......Page 326
12.2 The angle bisector......Page 327
12.3 The perpendicular bisector......Page 331
12.4 The altitudes......Page 332
12.5 Special lines of a triangle at a glance......Page 334
12.6 Special points in a triangle......Page 336
13.1 Definitions of special quadrilaterals......Page 337
13.2 Regular polygons......Page 339
13.3 The sum of the angles of a polygon......Page 340
14.1 Trapezoid......Page 343
Area of a trapezoid......Page 344
14.2 Parallelogram......Page 345
Area of a parallelogram......Page 346
14.3 Rectangle......Page 347
Area of a rhombus......Page 348
Area of a square......Page 351
15 Similarity......Page 352
15.1 Similar triangles......Page 353
15.2 Similarity of polygons and area of similar polygons......Page 356
15.3 A third proof of the Pythagorean theorem......Page 357
16.1 Circles passing through a point......Page 358
16.2 Circles passing through two points......Page 359
16.3 Circles passing through three points......Page 361
17.1 The relative positions of a circle and a line......Page 362
Circles tangent to one straight line......Page 363
Circles tangent to three straight lines......Page 365
18.1 The relative positions of two circles......Page 366
18.2 The relative positions of three circles......Page 370
19 Circles and angles......Page 372
19.1 Inscribed angles......Page 373
19.3 An angle with its vertex outside a circle......Page 379
Extreme positions of a circle and an angle......Page 381
19.4 An angle which a segment subtends......Page 384
20.1 Inscribed and circumscribed triangles......Page 390
20.2 Some exercises on inscribed and circumscribed triangles......Page 395
20.3 The area of a circumscribed triangle. The area of an inscribed triangle......Page 397
21.1 Inscribed polygons......Page 400
21.2 Inscribed quadrilaterals. Ptolemy’s theorem......Page 402
21.3 Some problems on inscribed quadrilaterals......Page 404
21.4 The relation between a circle and a regular polygon with n vertices......Page 407
22.1 Circumference......Page 410
22.2 The number π......Page 412
22.3 Length of an arc......Page 415
22.4 Radian measure of an angle......Page 416
23.1 Area of a regular polygon......Page 417
23.3 Area of a sector......Page 418
24 Overview of Chapter IV......Page 419
Glossary......Page 421