PROBLEMS IN PLANE AND SOLID GEOMETRY v.1 …

1 PROBLEMS IN PLANE AND PLANE GeometryViktor Prasolovtranslated and edited by Dimitry book has no equal. The priceless treasures of elementary GEOMETRY arenowhere else exposed in so complete and at the same time transparent form. The shortsolutions take barely 2 times more space than the formulations, while still remainingcomplete, with no gaps whatsoever, although many of the PROBLEMS are quite difficult. Onlythis enabled the author to squeeze about 2000 PROBLEMS on PLANE GEOMETRY in the book ofvolume of ca 600 pages thus embracing practically all the known PROBLEMS and theoremsofelementary book contains non-standard geometric PROBLEMS of a level higher than that of theproblems usually offered at high school.

2 The collection consists of two parts. Itis based onthree Russian editions of Prasolov s books on PLANE text is considerably modified for the English edition. Many new PROBLEMS are addedand detailed structuring in accordance with the methods of solution is book is addressed to high school students, teachers of mathematics, mathematicalclubs, and college s preface11 From the Author s preface12 Chapter 1. SIMILAR TRIANGLES15 Background15 Introductory problems15 1. Line segments intercepted by parallel lines15 2.

3 The ratio of sides of similar triangles17 3. The ratio of the areas of similar triangles18 4. Auxiliary equal triangles18* * *19 5. The triangle determined by the bases of the heights19 6. Similar figures20 PROBLEMS for independent study20 Solutions21 CHAPTER 2. INSCRIBED ANGLES33 Background33 Introductory problems33 1. Angles that subtend equal arcs34 2. The value of an angle between two chords35 3. The angle between a tangent and a chord35 4. Relations between the values of an angle and the lengths ofthe arc and chordassociated with the angle36 5.

4 Four points on one circle36 6. The inscribed angle and similar triangles37 7. The bisector divides an arc in halves38 8. An inscribed quadrilateral with perpendicular diagonals39 9. Three circumscribed circles intersect at one point39 10. Michel s point40 11. Miscellaneous problems40 PROBLEMS for independent study41 Solutions41 CHAPTER 3. CIRCLES57 Background57 Introductory problems58 1. The tangents to circles58 2. The product of the lengths of a chord s segments59 3. Tangent circles59 4. Three circles of the same radius60 5.

5 Two tangents drawn from one point6134 CONTENTS 61 6. Application of the theorem on triangle s heights61 7. Areas of curvilinear figures62 8. Circles inscribed in a disc segment62 9. Miscellaneous problems63 10. The radical axis63 PROBLEMS for independent study65 Solutions65 CHAPTER 4. AREA79 Background79 Introductory problems79 1. A median divides the triangleinto triangles of equal areas79 2. Calculation of areas80 3. The areas of the triangles into whicha quadrilateral is divided81 4. The areas of the parts into whicha quadrilateral is divided81 5.

6 Miscellaneous problems82* * *82 6. Lines and curves that divide figuresinto parts of equal area83 7. Formulas for the area of a quadrilateral83 8. An auxiliary area84 9. Regrouping areas85 PROBLEMS for independent study86 Solutions86 CHAPTER 5. TRIANGLES99 Background99 Introductory problems991. The inscribed and the circumscribed circles100* * *100* * *100 2. Right triangles101 3. The equilateral triangles101* * *101 4. Triangles with angles of 60 and 120 102 5. Integer triangles102 6. Miscellaneous problems103 7.

7 Menelaus s theorem104* * *105 8. Ceva s theorem106 9. Simson s line107 10. The pedal triangle108 11. Euler s line and the circle of nine points109 12. Brokar s points110 13. Lemoine s point111 CONTENTS5* * *111 PROBLEMS for independent study112 Solutions112 Chapter 6. POLYGONS137 Background137 Introductory problems137 1. The inscribed and circumscribed quadrilaterals137* * *138* * *138 2. Quadrilaterals139 3. Ptolemy s theorem140 4. Pentagons141 5. Hexagons141 6. Regular polygons142* * *142* * *143 7.

8 The inscribed and circumscribed polygons144* * *144 8. Arbitrary convex polygons144 9. Pascal s theorem145 PROBLEMS for independent study145 Solutions146 Chapter 7. LOCI169 Background169 Introductory problems169 1. The locus is a line or a segment of a line169* * *170 2. The locus is a circle or an arc of a circle170* * *170 3. The inscribed angle171 4. Auxiliary equal triangles171 5. The homothety171 6. A method of loci171 7. The locus with a nonzero area172 8. Carnot s theorem172 9. Fermat-Apollonius s circle173 PROBLEMS for independent study173 Solutions174 Chapter 8.

9 CONSTRUCTIONS183 1. The method of loci183 2. The inscribed angle183 3. Similar triangles and a homothety183 4. Construction of triangles from various elements183 5. Construction of triangles given various points184 6. Triangles184 7. Quadrilaterals185 8. Circles1856 CONTENTS 9. Apollonius circle186 10. Miscellaneous problems186 11. Unusual constructions186 12. Construction with a ruler only186 13. Constructions with the help of a two-sided ruler187 14. Constructions using a right angle188 PROBLEMS for independent study188 Solutions189 Chapter 9.

10 GEOMETRIC INEQUALITIES205 Background205 Introductory problems205 1. A median of a triangle205 2. Algebraic PROBLEMS on the triangle inequality206 3. The sum of the lengths of quadrilateral s diagonals206 4. Miscellaneous PROBLEMS on the triangle inequality207* * *207 5. The area of a triangle does not exceed a half product of two sides207 6. Inequalities of areas208 7. Area. One figure lies inside another209* * *209 8. Broken lines inside a square209 9. The quadrilateral210 10. Polygons210* * *211 11.