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 exp osed 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, with no gaps whatsoever, although many of the problems are quite difficult. 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. The text is considerably modified for the English edition. Many new problems are added 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 p olygons 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 produ ct 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 [...]... problems were added, particularly those concerning the geometry of the triangle I was greatly influenced in the process by the second edition of the book by I F Sharygin Problems on Geometry Plane geometry, Nauka, Moscow,1986 and a wonderful and undeservedly forgotten book by D Efremov New Geometry of the Triangle, Matezis, Odessa, 1902 The present book can be used not only as a source of optional problems... problems, an enormity too vast to grasp Second, even if this might have been possible, the book would be terribly overloaded, and therefore of no interest to anybody However, in the book Problems in plane geometry followed by Problems in solid geometry this task is successfully perfomed In the process of writing the book the author used the books and magazines published in the last century as well as modern... theorems of elementary geometry was considered too wide to grasp in full even in the last century Even nowadays the stream of new problems is still wide (The majority of these problems, however, are either well-forgotten old ones or those recently pirated from a neighbouring country.) Any attempt to collect an encyclopedia of all the problems seems to be doomed to failure for many reasons First of all,... statement desired 1.61 Let M = A Then XA = A; hence, AYA = 1 Similarly, CXC = 1 Let us prove that y = AYA and x = CXC are the desired lines On side BC, take point D so that AB M D, see Fig 11 Let E be the intersection point of lines CXC and M D Then, XM M + YM M = XC E + YM M Since △ABC ∼ △M DC, it follows that CE = YM M Therefore, CE = YM M Hence, XM M + YM M = XC E + CE = XC C = 1 Figure 11 (Sol 1.61)... C is being replaced by another triangle BM AM similar to the initial one Therefore, the quantity CM remains a constant Analogously, DM remains a constant 1.27 Let medians meet at O; denote the intersection points of median AK with lines F P and F E by Q and M , respectively; denote the intersection points of median CL with lines EP and F E by R and N , respectively (Fig 5) Clearly, F M : F E = F Q :... circulation, the total over 1 000 000 copies The 3rd edition of Problems in Plane Geometry was issued in 1996 and the latest one in 2001 The readers’ interest is partly occasioned by a well-thought classification system The collection consists of three parts Part 1 covers classical subjects of plane geometry It contains nearly 1000 problems with complete solutions and over 100 problems to be solved on one’s... more suitable for contests and for use in mathematical clubs The problems cover cuttings, colorings, the pigeonhole (or Dirichlet’s) principle, induction, and so on Part 3 is devoted to solid geometry A rather detailed table of contents serves as a guide in the sea of geometric problems It helps the experts to easily find what they need while the uninitiated can quickly learn what exactly is that they... might be somewhat depressingly overwhelming Advice and comments given by Academician A V Pogorelov, and Professors A M Abramov, A Yu Vaintrob, N B Vasiliev, N P Dolbilin, and S Yu Orevkov were a great help to me in preparing the first Soviet edition I wish to express my sincere gratitude to all of them To save space, sections with background only contain the material directly pertinent to the respective... ABCD beyond A and C, respectively Line segment KL intersects sides AB and CD at M and N , respectively; KL intersects diagonals AC and BD at O and P , respectively Prove that if KM = N L, then KO = P L 1.16 Points P , Q, R, and S on sides AB, BC, CD and DA, respectively, of convex quadrilateral ABCD are such that BP : AB = CR : CD = α and AS : AD = BQ : BC = β Prove that P R and QS are divided by their... best suited for coaching for a mathematical Olympiad than for a regular class work: the level of difficulty increases rather fast Problems in each section are ordered difficulty-wise The first problems of the sections are simple; they are a match for many Here are some examples: 1Here are a few samples: v 96, n 5, 1989, p 429–431 (here the main idea of the solution is the right illustration — precisely the . 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 articularly those concern ing the ge- ometry of the triangle. I was greatly influenced in the process by the second edition of the book by I. F. Sharygin Problems on Geometry. Plane geometry, Nauka,. 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