Problems in Plane and Solid Geometry, Volume 1, Plane Geometry

Title......Page 1
Abstract......Page 2
Contents......Page 3
Editor’s preface......Page 11
§1. Line segments intercepted by parallel lines......Page 15
§2. The ratio of sides of similar trian......Page 17
§4. Auxiliary equal triangles......Page 18
§5. The triangle determined by the bases of the heights......Page 19
Problems for independent study......Page 20
Solutions......Page 21
Introductory problems......Page 33
§1. Angles that subtend equal arcs......Page 34
§3. The angle between a tangent and a chord......Page 35
§5. Four points on one circle......Page 36
§6. The inscribed angle and similar triangles......Page 37
§7. The bisector divides an arc in halves......Page 38
§9. Three circumscribed circles intersect at one point......Page 39
§11. Miscellaneous problem......Page 40
Solutions......Page 41
Background......Page 57
§1. The tangents to circles......Page 58
§3. Tangent circles......Page 59
§4. Three circles of the same radius......Page 60
§6. Application of the theorem on triangle’s heights......Page 61
§8. Circles inscribed in a disc segment......Page 62
§10. The radical axis......Page 63
Solutions......Page 65
§1. A median divides the triangle into triangles of equal areas......Page 79
§2. Calculation of areas......Page 80
§4. The areas of the parts into which a quadrilateral is divided......Page 81
§5. Miscellaneous problems......Page 82
§7. Formulas for the area of a quadrilateral......Page 83
§8. An auxiliary area......Page 84
§9. Regrouping areas......Page 85
Solutions......Page 86
Introductory problems......Page 99
1. The inscribed and the circumscribed circles......Page 100
§3. The equilateral triangles......Page 101
§5. Integer triangles......Page 102
§6. Miscellaneous problems......Page 103
§7. Menelaus’s theorem......Page 104
§8. Ceva’s theorem......Page 106
§9. Simson’s line......Page 107
§10. The pedal triangle......Page 108
§11. Euler’s line and the circle of nine points......Page 109
§12. Brokar’s points......Page 110
§13. Lemoine’s point......Page 111
Solutions......Page 112
§1. The inscribed and circumscribed quadrilaterals......Page 137
§2. Quadrilaterals......Page 139
§3. Ptolemy’s theorem......Page 140
§5. Hexagons......Page 141
§6. Regular polygons......Page 142
§8. Arbitrary convex polygons......Page 144
Problems for independent study......Page 145
Solutions......Page 146
§1. The locus is a line or a segment of a line......Page 169
§2. The locus is a circle or an arc of a circle......Page 170
§6. A method of loci......Page 171
§8. Carnot’s theorem......Page 172
Problems for independent study......Page 173
Solutions......Page 174
§4. Construction of triangles from various elements......Page 183
§6. Triangles......Page 184
§8. Circles......Page 185
§12. Construction with a ruler only......Page 186
§13. Constructions with the help of a two-sided ruler......Page 187
Problems for independent study......Page 188
Solutions......Page 189
§1. A median of a triangle......Page 205
§3. The sum of the lengths of quadrilateral’s diagonals......Page 206
§5. The area of a triangle does not exceed a half product of two sides......Page 207
§6. Inequalities of areas......Page 208
§8. Broken lines inside a square......Page 209
§10. Polygons......Page 210
§11. Miscellaneous problems......Page 211
Supplement. Certain inequalities......Page 212
Solutions......Page 213
§3. The bisectors......Page 235
§6. Symmetric inequalities between the angles of a triangle......Page 236
§8. Inequalities for the area of a triangle......Page 237
§11. Inequalities for right triangles......Page 238
§13. Inequalities in triangles......Page 239
Solutions......Page 240
§1. The triangle......Page 255
§2. Extremal points of a triangle......Page 256
§5. Polygons......Page 257
Problems for independent study......Page 258
Solutions......Page 259
§1. The law of sines......Page 271
§3. The inscribed, the circumscribed and escribed circles; their radii......Page 272
§5. The sines and cosines of a triangle’s angles......Page 273
§7. Calculation of angles......Page 274
§9. Miscellaneous problems......Page 275
§10. The method of coordinates......Page 276
Solutions......Page 277
Background......Page 289
§2. Inner product. Relations......Page 290
§3. Inequalities......Page 291
§5. Auxiliary projections......Page 292
§7. Pseudoinner product......Page 293
Problems for independent study......Page 294
Solutions......Page 295
§1. Main properties of the center of mass......Page 307
§2. A theorem on mass regroupping......Page 308
§3. The moment of inertia......Page 309
§5. The barycentric coordinates......Page 310
Solutions......Page 311
§1. Solving problems with the aid of parallel translations......Page 319
Solutions......Page 320
§1. Solving problems with the help of a symmetry......Page 327
§3. Solving problems with the help of a symmetry. Constructions......Page 328
Solutions......Page 329
§1. Solving problems with the help of a symmetry......Page 335
§4. Compositions of symmetries......Page 336
§6. Chasles’s theorem......Page 337
Solutions......Page 338
§1. Rotation by 90◦......Page 345
§2. Rotation by 60◦......Page 346
§4. Compositions of rotations......Page 347
Problems for independent study......Page 348
Solutions......Page 349
§1. Homothetic polygons......Page 359
§3. Costructions and loci......Page 360
§5. Rotational homothety......Page 361
§6. The center of a rotational homothety......Page 362
§7. The similarity circle of three figures......Page 363
Solutions......Page 364
§1. The least and the greatest angles......Page 375
§5. The convex hull and the base lines......Page 376
Solutions......Page 378
§1. The case when there are finitely many points, lines, etc.......Page 385
§2. Angles and lengths......Page 386
Solutions......Page 387
§1. Convex polygons......Page 397
§3. Non-convex polygons......Page 398
Solutions......Page 399
§1. Even and odd......Page 409
§3. Invariants......Page 410
§4. Auxiliary colorings......Page 411
§6. Problems on colorings......Page 412
Solutions......Page 413
§2. Miscellaneous problems......Page 425
Solutions......Page 426
§2. How lines cut the plane......Page 431
Solutions......Page 432
§2. Systems of segments, lines and circles......Page 437
Solutions......Page 438
Solutions......Page 445
§1. Properties of inversions......Page 449
§3. Constructions with the help of a compass only......Page 450
§4. Let us perform an inversion......Page 451
§5. Points that lie on one circle and circles passing through one point......Page 452
§6. Chains of circles......Page 454
Solutions......Page 455
§1. Affine transformations......Page 465
Solutions......Page 466
§1. Projective transformations of the line......Page 473
§2. Projective transformations of the plane......Page 474
§3. Let us transform the given line into the infinite one......Page 477
§4. Application of projective maps that preserve a circle......Page 478
§6. Application of projective transformations of the line in problems on construction......Page 479
Solutions......Page 480
Index......Page 493