planegeo - PROBLEMS IN PLANE AND SOLID GEOMETRY v.1 Plane...

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Unformatted text preview: 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...
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This note was uploaded on 07/04/2009 for the course MATH 99 taught by Professor Laoshi during the Spring '09 term at Fudan University.

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planegeo - PROBLEMS IN PLANE AND SOLID GEOMETRY v.1 Plane...

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