PROBLEMS IN PLANE AND SOLID GEOMETRY v.1 Plane Geometry Viktor Prasolov translated ...

495 Pages·2005·3.95 MB·English

PROBLEMS IN PLANE AND SOLID
GEOMETRY
v.1 Plane Geometry
Viktor Prasolov
translated and edited by Dimitry Leites
Abstract. This book has no equal. The priceless treasures of elementary geometry are
nowhere else exposed in so complete and at the same time transparent form. The short
solutions take barely 1.5 2 times more space than the formulations, while still remaining
−
complete, withnogapswhatsoever, althoughmanyoftheproblemsarequitedifficult. Only
this enabled the author to squeeze about 2000 problems on plane geometry in the book of
volume of ca 600 pages thus embracing practically all the known problems and theorems of
elementary geometry.
The book contains non-standard geometric problems of a level higher than that of the
problems usually offered at high school. The collection consists of two parts. It is based on
three Russian editions of Prasolov’s books on plane geometry.
ThetextisconsiderablymodifiedfortheEnglishedition. Manynewproblemsareadded
and detailed structuring in accordance with the methods of solution is adopted.
The book is addressed to high school students, teachers of mathematics, mathematical
clubs, and college students.
Contents
Editor’s preface 11
From the Author’s preface 12
Chapter 1. SIMILAR TRIANGLES 15
Background 15
Introductory problems 15
1. Line segments intercepted by parallel lines 15
§
2. The ratio of sides of similar triangles 17
§
3. The ratio of the areas of similar triangles 18
§
4. Auxiliary equal triangles 18
§
* * * 19
5. The triangle determined by the bases of the heights 19
§
6. Similar figures 20
§
Problems for independent study 20
Solutions 21
CHAPTER 2. INSCRIBED ANGLES 33
Background 33
Introductory problems 33
1. Angles that subtend equal arcs 34
§
2. The value of an angle between two chords 35
§
3. The angle between a tangent and a chord 35
§
4. Relations between the values of an angle and the lengths of the arc and chord
§
associated with the angle 36
5. Four points on one circle 36
§
6. The inscribed angle and similar triangles 37
§
7. The bisector divides an arc in halves 38
§
8. An inscribed quadrilateral with perpendicular diagonals 39
§
9. Three circumscribed circles intersect at one point 39
§
10. Michel’s point 40
§
11. Miscellaneous problems 40
§
Problems for independent study 41
Solutions 41
CHAPTER 3. CIRCLES 57
Background 57
Introductory problems 58
1. The tangents to circles 58
§
2. The product of the lengths of a chord’s segments 59
§
3. Tangent circles 59
§
4. Three circles of the same radius 60
§
5. Two tangents drawn from one point 61
§
3
4 CONTENTS
61
∗∗∗
6. Application of the theorem on triangle’s heights 61
§
7. Areas of curvilinear figures 62
§
8. Circles inscribed in a disc segment 62
§
9. Miscellaneous problems 63
§
10. The radical axis 63
§
Problems for independent study 65
Solutions 65
CHAPTER 4. AREA 79
Background 79
Introductory problems 79
1. A median divides the triangle
§
into triangles of equal areas 79
2. Calculation of areas 80
§
3. The areas of the triangles into which
§
a quadrilateral is divided 81
4. The areas of the parts into which
§
a quadrilateral is divided 81
5. Miscellaneous problems 82
§
* * * 82
6. Lines and curves that divide figures
§
into parts of equal area 83
7. Formulas for the area of a quadrilateral 83
§
8. An auxiliary area 84
§
9. Regrouping areas 85
§
Problems for independent study 86
Solutions 86
CHAPTER 5. TRIANGLES 99
Background 99
Introductory problems 99
1. The inscribed and the circumscribed circles 100
* * * 100
* * * 100
2. Right triangles 101
§
3. The equilateral triangles 101
§
* * * 101
4. Triangles with angles of 60 and 120 102
◦ ◦
§
5. Integer triangles 102
§
6. Miscellaneous problems 103
§
7. Menelaus’s theorem 104
§
* * * 105
8. Ceva’s theorem 106
§
9. Simson’s line 107
§
10. The pedal triangle 108
§
11. Euler’s line and the circle of nine points 109
§
12. Brokar’s points 110
§
13. Lemoine’s point 111
§
CONTENTS 5
* * * 111
Problems for independent study 112
Solutions 112
Chapter 6. POLYGONS 137
Background 137
Introductory problems 137
1. The inscribed and circumscribed quadrilaterals 137
§
* * * 138
* * * 138
2. Quadrilaterals 139
§
3. Ptolemy’s theorem 140
§
4. Pentagons 141
§
5. Hexagons 141
§
6. Regular polygons 142
§
* * * 142
* * * 143
7. The inscribed and circumscribed polygons 144
§
* * * 144
8. Arbitrary convex polygons 144
§
9. Pascal’s theorem 145
§
Problems for independent study 145
Solutions 146
Chapter 7. LOCI 169
Background 169
Introductory problems 169
1. The locus is a line or a segment of a line 169
§
* * * 170
2. The locus is a circle or an arc of a circle 170
§
* * * 170
3. The inscribed angle 171
§
4. Auxiliary equal triangles 171
§
5. The homothety 171
§
6. A method of loci 171
§
7. The locus with a nonzero area 172
§
8. Carnot’s theorem 172
§
9. Fermat-Apollonius’s circle 173
§
Problems for independent study 173
Solutions 174
Chapter 8. CONSTRUCTIONS 183
1. The method of loci 183
§
2. The inscribed angle 183
§
3. Similar triangles and a homothety 183
§
4. Construction of triangles from various elements 183
§
5. Construction of triangles given various points 184
§
6. Triangles 184
§
7. Quadrilaterals 185
§
8. Circles 185
§
6 CONTENTS
9. Apollonius’ circle 186
§
10. Miscellaneous problems 186
§
11. Unusual constructions 186
§
12. Construction with a ruler only 186
§
13. Constructions with the help of a two-sided ruler 187
§
14. Constructions using a right angle 188
§
Problems for independent study 188
Solutions 189
Chapter 9. GEOMETRIC INEQUALITIES 205
Background 205
Introductory problems 205
1. A median of a triangle 205
§
2. Algebraic problems on the triangle inequality 206
§
3. The sum of the lengths of quadrilateral’s diagonals 206
§
4. Miscellaneous problems on the triangle inequality 207
§
* * * 207
5. The area of a triangle does not exceed a half product of two sides 207
§
6. Inequalities of areas 208
§
7. Area. One figure lies inside another 209
§
* * * 209
8. Broken lines inside a square 209
§
9. The quadrilateral 210
§
10. Polygons 210
§
* * * 211
11. Miscellaneous problems 211
§
* * * 211
Problems for independent study 212
Supplement. Certain inequalities 212
Solutions 213
Chapter 10. INEQUALITIES BETWEEN THE ELEMENTS OF A TRIANGLE 235
1. Medians 235
§
2. Heights 235
§
3. The bisectors 235
§
4. The lengths of sides 236
§
5. The radii of the circumscribed, inscribed and escribed circles 236
§
6. Symmetric inequalities between the angles of a triangle 236
§
7. Inequalities between the angles of a triangle 237
§
8. Inequalities for the area of a triangle 237
§
* * * 238
9. The greater angle subtends the longer side 238
§
10. Any segment inside a triangle is shorter than the longest side 238
§
11. Inequalities for right triangles 238
§
12. Inequalities for acute triangles 239
§
13. Inequalities in triangles 239
§
Problems for independent study 240
Solutions 240
Chapter 11. PROBLEMS ON MAXIMUM AND MINIMUM 255
CONTENTS 7
Background 255
Introductory problems 255
1. The triangle 255
§
* * * 256
2. Extremal points of a triangle 256
§
3. The angle 257
§
4. The quadrilateral 257
§
5. Polygons 257
§
6. Miscellaneous problems 258
§
7. The extremal properties of regular polygons 258
§
Problems for independent study 258
Solutions 259
Chapter 12. CALCULATIONS AND METRIC RELATIONS 271
Introductory problems 271
1. The law of sines 271
§
2. The law of cosines 272
§
3. The inscribed, the circumscribed and escribed circles; their radii 272
§
4. The lengths of the sides, heights, bisectors 273
§
5. The sines and cosines of a triangle’s angles 273
§
6. The tangents and cotangents of a triangle’s angles 274
§
7. Calculation of angles 274
§
* * * 274
8. The circles 275
§
* * * 275
9. Miscellaneous problems 275
§
10. The method of coordinates 276
§
Problems for independent study 277
Solutions 277
Chapter 13. VECTORS 289
Background 289
Introductory problems 289
1. Vectors formed by polygons’ (?) sides 290
§
2. Inner product. Relations 290
§
3. Inequalities 291
§
4. Sums of vectors 292
§
5. Auxiliary projections 292
§
6. The method of averaging 293
§
7. Pseudoinner product 293
§
Problems for independent study 294
Solutions 295
Chapter 14. THE CENTER OF MASS 307
Background 307
1. Main properties of the center of mass 307
§
2. A theorem on mass regroupping 308
§
3. The moment of inertia 309
§
4. Miscellaneous problems 310
§
5. The barycentric coordinates 310
§
8 CONTENTS
Solutions 311
Chapter 15. PARALLEL TRANSLATIONS 319
Background 319
Introductory problems 319
1. Solving problems with the aid of parallel translations 319
§
2. Problems on construction and loci 320
§
* * * 320
Problems for independent study 320
Solutions 320
Chapter 16. CENTRAL SYMMETRY 327
Background 327
Introductory problems 327
1. Solving problems with the help of a symmetry 327
§
2. Properties of the symmetry 328
§
3. Solving problems with the help of a symmetry. Constructions 328
§
Problems for independent study 329
Solutions 329
Chapter 17. THE SYMMETRY THROUGH A LINE 335
Background 335
Introductory problems 335
1. Solving problems with the help of a symmetry 335
§
2. Constructions 336
§
* * * 336
3. Inequalities and extremals 336
§
4. Compositions of symmetries 336
§
5. Properties of symmetries and axes of symmetries 337
§
6. Chasles’s theorem 337
§
Problems for independent study 338
Solutions 338
Chapter 18. ROTATIONS 345
Background 345
Introductory problems 345
1. Rotation by 90 345
◦
§
2. Rotation by 60 346
◦
§
3. Rotations through arbitrary angles 347
§
4. Compositions of rotations 347
§
* * * 348
* * * 348
Problems for independent study 348
Solutions 349
Chapter 19. HOMOTHETY AND ROTATIONAL HOMOTHETY 359
Background 359
Introductory problems 359
1. Homothetic polygons 359
§
2. Homothetic circles 360
§
3. Costructions and loci 360
§
CONTENTS 9
* * * 361
4. Composition of homotheties 361
§
5. Rotational homothety 361
§
* * * 362
* * * 362
6. The center of a rotational homothety 362
§
7. The similarity circle of three figures 363
§
Problems for independent study 364
Solutions 364
Chapter 20. THE PRINCIPLE OF AN EXTREMAL ELEMENT 375
Background 375
1. The least and the greatest angles 375
§
2. The least and the greatest distances 376
§
3. The least and the greatest areas 376
§
4. The greatest triangle 376
§
5. The convex hull and the base lines 376
§
6. Miscellaneous problems 378
§
Solutions 378
Chapter 21. DIRICHLET’S PRINCIPLE 385
Background 385
1. The case when there are finitely many points, lines, etc. 385
§
2. Angles and lengths 386
§
3. Area 387
§
Solutions 387
Chapter 22. CONVEX AND NONCONVEX POLYGONS 397
Background 397
1. Convex polygons 397
§
* * * 397
2. Helly’s theorem 398
§
3. Non-convex polygons 398
§
Solutions 399
Chapter 23. DIVISIBILITY, INVARIANTS, COLORINGS 409
Background 409
1. Even and odd 409
§
2. Divisibility 410
§
3. Invariants 410
§
4. Auxiliary colorings 411
§
5. More auxiliary colorings 412
§
* * * 412
6. Problems on colorings 412
§
* * * 413
Solutions 413
Chapter 24. INTEGER LATTICES 425
1. Polygons with vertices in the nodes of a lattice 425
§
2. Miscellaneous problems 425
§
Solutions 426
10 CONTENTS
Chapter 25. CUTTINGS 431
1. Cuttings into parallelograms 431
§
2. How lines cut the plane 431
§
Solutions 432
Chapter 26. SYSTEMS OF POINTS AND SEGMENTS.
EXAMPLES AND COUNTEREXAMPLES 437
1. Systems of points 437
§
2. Systems of segments, lines and circles 437
§
3. Examples and counterexamples 438
§
Solutions 438
Chapter 27. INDUCTION AND COMBINATORICS 445
1. Induction 445
§
2. Combinatorics 445
§
Solutions 445
Chapter 28. INVERSION 449
Background 449
1. Properties of inversions 449
§
2. Construction of circles 450
§
3. Constructions with the help of a compass only 450
§
4. Let us perform an inversion 451
§
5. Points that lie on one circle and circles passing through one point 452
§
6. Chains of circles 454
§
Solutions 455
Chapter 29. AFFINE TRANSFORMATIONS 465
1. Affine transformations 465
§
2. How to solve problems with the help of affine transformations 466
§
Solutions 466
Chapter 30. PROJECTIVE TRANSFORMATIONS 473
1. Projective transformations of the line 473
§
2. Projective transformations of the plane 474
§
3. Let us transform the given line into the infinite one 477
§
4. Application of projective maps that preserve a circle 478
§
5. Application of projective transformations of the line 479
§
6. Application of projective transformations of the line in problems on construction 479
§
7. Impossibility of construction with the help of a ruler only 480
§
Solutions 480
Index 493