Geometry

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