А.В.Погорелов. Геометрия 7-11 сынып, 2002

Chapter I. Plane geometry

§ 1. Basic properties of simple geometric figures

1. Geometric figures

2. Point and line

3. Segment

4. Measuring segments

5. Half-planes

6. Ray

7. Angle

8. Constructing segments and angles by measurement

9. Triangle

10. Existence of a triangle congruent to a given triangle

11. Parallel lines

12. Theorems and proofs

13. Axioms

Review questions

Problems

§ 2. Adjacent and vertical angles

14. Adjacent angles

15. Vertical angles

16. Perpendicular lines

17. Proof by contradiction

18. Angle bisector

19. What to do to succeed in geometry

Review questions

Problems

§ 3. Criteria for congruence of triangles

20. First criterion for congruence of triangles

21. Using axioms in proving theorems

22. Second criterion for congruence of triangles

23. Isosceles triangle

24. Converse theorem

25. Altitude, bisector and median of a triangle

26. Property of the median of an isosceles triangle

27. Third criterion for congruence of triangles

28. How to study independently using the textbook

Review questions

Problems

§ 4. Sum of the angles of a triangle

29. Parallelism of lines

30. Angles formed by a transversal cutting two lines

31. Criterion for parallel lines

32. Property of angles formed by a transversal of parallel lines

33. Sum of the angles of a triangle

34. Exterior angles of a triangle

35. Right triangle

36. Existence and uniqueness of a perpendicular to a line

37. From the history of the origin of geometry

Review questions

Problems

§ 5. Geometric constructions

38. Circle

39. Circumcircle of a triangle

40. Tangent to a circle

41. Incircle of a triangle

42. What construction problems are

43. Constructing a triangle from given sides

44. Constructing an angle equal to a given angle

45. Constructing an angle bisector

46. Bisecting a segment

47. Constructing a perpendicular line

48. Locus of points

49. Method of loci

Review questions

Problems

§ 6. Quadrilaterals

50. Definition of a quadrilateral

51. Parallelogram

52. Property of the diagonals of a parallelogram

53. Property of opposite sides and angles of a parallelogram

54. Rectangle

55. Rhombus

56. Square

57. Thales' theorem

58. Midline of a triangle

59. Trapezoid

60. Theorem on proportional segments

61. Constructing the fourth proportional segment

Review questions

Problems

§ 7. Pythagorean theorem

62. Cosine of an angle

63. Pythagorean theorem

64. Egyptian triangle

65. Perpendicular and oblique line

66. Triangle inequality

67. Relations between sides and angles of a right triangle

68. Fundamental trigonometric identity

69. Finding sine and tangent from cosine

70. Finding values of sine, cosine and tangent from an angle

Review questions

Problems

§ 8. Cartesian coordinates in the plane

71. Definition of Cartesian coordinates

72. Coordinates of the midpoint of a segment

73. Distance between points

74. Equation of a circle

75. Equation of a line

76. Coordinates of the intersection point of lines

77. Position of a line relative to the coordinate system

78. Slope coefficient of a line equation

79. Graph of a linear function

80. Intersection of a line and a circle

81. Defining sine, cosine and tangent for any angle from 0° to 180°

Review questions

Problems

§ 9. Motion

82. Transformation of figures

83. Properties of motion

84. Symmetry with respect to a point

85. Symmetry with respect to a line

86. Rotation

87. Translation and its properties

88. Existence and uniqueness of translation

89. Directions of rays

90. Congruence of figures

Review questions

Problems

§ 10. Vectors

91. Magnitude and direction of a vector

92. Equality of vectors

93. Coordinates of a vector

94. Addition of vectors

95. Addition of forces

96. Multiplication of a vector by a number

97. Decomposition of a vector by two non-collinear vectors

98. Dot product of vectors

99. Decomposition of a vector along coordinate axes

Review questions

Problems

§ 11. Similarity of figures

100. Similarity transformation

101. Properties of a similarity transformation

102. Similarity of figures

103. Similarity criterion for triangles by two angles

104. Similarity criterion for triangles by two sides and the included angle

105. Similarity criterion for triangles by three sides

106. Similarity of right triangles

107. Inscribed angles

108. Proportional segments of chords and secants of a circle

Review questions

Problems

§ 12. Solving triangles

109. Law of cosines

110. Law of sines

111. Relation between angles of a triangle and opposite sides

112. Solving triangles

Review questions

Problems

§ 13. Polygons

113. Broken line

114. Convex polygons

115. Regular polygons

116. Formulas for radii of incircles and circumcircles of regular polygons

117. Constructing some regular polygons

118. Properties of a regular hexagon

119. Circumference

120. Radian measure of an angle

Review questions

Problems

§ 14. Areas of figures

121. Concept of area

122. Area of a rectangle

123. Area of a parallelogram

124. Area of a triangle

125. Heron's formula for the area of a triangle

126. Area of a trapezoid

127. Formulas for radii of the incircle and circumcircle of a triangle

128. Areas of similar figures

129. Area of a disk

Review questions

Problems

§ 15. Axioms of solid geometry and their simplest consequences

130. Axioms of solid geometry

131. Existence of a plane through a given line and a given point

132. Intersection of a line and a plane

133. Existence of a plane through three given points

134. Remark on Axiom I

135. Dividing space by a plane into two half-spaces

Review questions

Problems

§ 16. Parallelism of lines and planes

136. Parallel lines in space

137. Criterion for parallel lines

138. Criterion for parallelism of a line and a plane

139. Criterion for parallel planes

140. Existence of a plane parallel to a given plane

141. Properties of parallel planes

142. Plane sections of spatial figures

Review questions

Problems

§ 17. Perpendicularity of lines and planes

143. Perpendicularity of lines in space

144. Criterion for perpendicularity of a line and a plane

145. Constructing a perpendicular line and plane

146. Properties of a perpendicular line and plane

147. Perpendicular and oblique line

148. Theorem of three perpendiculars

149. Criterion for perpendicular planes

150. Distance between skew lines

151. Orthogonal projection and its application in technical drawing

Review questions

Problems

§ 18. Cartesian coordinates and vectors in space

152. Introducing Cartesian coordinates in space

153. Distance between points

154. Coordinates of the midpoint of a segment

155. Transformation by symmetry in space

156. Symmetry in nature and technology

157. Motion in space

158. Translation in space

159. Similarity of spatial figures

160. Angles between skew lines

161. Angle between a line and a plane

162. Angle between planes

163. Area of the orthogonal projection of a polygon

164. Vectors in space

165. Operations on vectors in space

Review questions

Problems

§ 19. Polyhedra

166. Dihedral angle

167. Trihedral and polyhedral angles

168. Polyhedron

169. Prism

170. Representing a prism and constructing its section

171. Right prism

172. Parallelepiped

173. Central symmetry of a parallelepiped

174. Rectangular parallelepiped

175. Symmetry of a rectangular parallelepiped

176. Pyramid

177. Constructing pyramids and their plane sections

178. Truncated pyramid

179. Regular pyramid

180. Regular polyhedra

Review questions

Problems

§ 20. Solids of revolution

181. Cylinder

182. Plane sections of a cylinder

183. Inscribed and circumscribed prisms

184. Cone

185. Plane sections of a cone

186. Inscribed and circumscribed pyramids

187. Ball

188. Plane section of a ball

189. Symmetry of a ball

190. Tangent plane to a ball

191. Intersection of two spheres

192. Inscribed and circumscribed polyhedra

193. Concept of a geometric solid and its surface

Review questions

Problems

§ 21. Volumes of polyhedra

194. Concept of volume

195. Volume of a rectangular parallelepiped

196. Volume of an oblique parallelepiped

197. Volume of a pyramid

198. Equal-volume solids

199. Volume of a pyramid

200. Volume of a truncated pyramid

201. Volumes of similar solids

Review questions

Problems

§ 22. Surfaces and volumes of solids of revolution

202. Volume of a cylinder

203. Volume of a cone

204. Volume of a truncated cone

205. General formula for volumes of solids of revolution

206. Volume of a ball

207. Volume of a spherical segment and sector

208. Lateral surface area of a cylinder

209. Lateral surface area of a cone

210. Surface area of a sphere

Review questions

Problems

Answers and hints to problems

Name index

Bibliography