Problems in plane and solid geometry v1. Plane geometry - V.Prasolov.pdf

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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 di±cult. 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 o®ered 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
§3. The ratio of the areas of similar triangles
17
§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
§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
§2. The ratio of sides of similar triangles
18
§7. The bisector divides an arc in halves
38
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
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
§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
§9. Regrouping areas
85
* * *
100
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
§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
* * *
138

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