BỘ GIÁO DỤC VÀ ĐÀO TẠO Tuyển tập các bài hình học vô địch thế giới P1 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, 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 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 circumscrib ed, 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 [...]... the extensions of sides AB and CD (Fig 12) Denote by M1 and P1 the other intersection points of lines RT and QP with the circle Figure 12 (Sol 1.65) Since T M1 = RM = AQ and T M1 AQ, it follows that AM1 T Q Similarly, AP1 RP Since ∠M1 AP1 = 90◦ , it follows that RP ⊥ T Q Denote the intersection points of lines T Q and RP , M1 A and RP , P1 A and T Q by E, F , G, respectively To prove that point E... O2 at points P1 and P2 , respectively We have to prove that P1 = P2 Let us consider points A, D1 , D2 — the intersection points of KL with M N , KM with O1 A, and LN with O2 A, respectively Since ∠O1 AM + ∠N AO2 = 90◦ , 32 CHAPTER 1 SIMILAR TRIANGLES right triangles O1 M A and AN O2 are similar; we also see that AO2 KM and AO1 LN Since these lines are parallel, AD1 : D1 O1 = O2 P1 : P1 O1 and D2... these lines are parallel, AD1 : D1 O1 = O2 P1 : P1 O1 and D2 O2 : AD2 = O2 P2 : P2 O1 The similarity of quadrilaterals AKO1 M and O2 N AL yields AD1 : D1 O1 = D2 O2 : AD2 Therefore, O2 P1 : P1 O1 = O2 P2 : P2 O1 , i.e., P1 = P2 CHAPTER 2 INSCRIBED ANGLES Background 1 Angle ∠ABC whose vertex lies on a circle and legs intersect this circle is called inscribed in the circle Let O be the center of the circle... To prove that point E lies on line AC, it suffices to prove that rectangles AF EG and AM1 CP1 are similar Since ∠ARF = ∠AM1 R = ∠M1 T G = ∠M1 CT , we may denote the values of these angles by the same letter α We have: AF = RA sin α = M1 A sin2 α and AG = M1 T sin α = M1 C sin2 α Therefore, rectangles AF EG and AM1 CP1 are similar 1.66 Denote the centers of the circles by O1 and O2 The outer tangent is... the sum of distances from point A to the tangent points of the inscribed circle with sides AB and AC; therefore, the sum of perimeters of small triangles is equal to the perimeter of triangle ABC, i.e., P1 + P2 + P3 = P The similarity of triangles implies that rri = Pi P Summing these equalities for all the i we get the statement desired 1.61 Let M = A Then XA = A; hence, AYA = 1 Similarly, CXC = 1 . GIÁO DỤC VÀ ĐÀO TẠO Tuyển tập các bài hình học vô địch thế giới P1 PROBLEMS IN PLANE AND SOLID GEOMETRY v.1