Preface
Symbol index
Contents
Part I Euclidean Geometry of the plane
Chapter 1 The basic notions
1.1 Undefined terms, axioms
1.2 Line and line segment
1.3 Length, distance
1.4 Angles
1.5 Angle kinds
1.6 Triangles
1.7 Congruence, the equality of shapes
1.8 Isosceles and right triangle
1.9 Triangle congruence criteria
1.10 Triangle’s sides and angles relations
1.11 The triangle inequality
1.12 The orthogonal to a line
1.13 The parallel from a point
1.14 The sum of triangle’s angles
1.15 The axiom of parallels
1.16 Symmetries
1.17 Ratios, harmonic quadruples
1.18 Comments and exercises for the chapter
References
Chapter 2 Circle and polygons
2.1 The circle, the diameter, the chord
2.2 Circle and line
2.3 Two circles
2.4 Constructions using ruler and compass
2.5 Parallelograms
2.6 Quadrilaterals
2.7 The middles of sides
2.8 The triangle’s medians
2.9 The rectangle and the square
2.10 Other kinds of quadrilaterals
2.11 Polygons, regular polygons
2.12 Arcs, central angles
2.13 Inscribed angles
2.14 Inscriptible or cyclic quadrilaterals
2.15 Circumscribed quadrilaterals
2.16 Geometric loci
2.17 Comments and exercises for the chapter
References
Chapter 3 Areas, Thales, Pythagoras, Pappus
3.1 Area of polygons
3.2 The area of the rectangle
3.3 Area of parallelogram and triangle
3.4 Pythagoras and Pappus
3.5 Similar right triangles
3.6 The trigonometric functions
3.7 The theorem of Thales
3.8 Pencils of lines
3.9 Similar triangles
3.10 Similar polygons
3.11 Triangle’s sine and cosine rules
3.12 Stewart, medians, bisectors, altitudes
3.13 Antiparallels, symmedians
3.14 Comments and exercises for the chapter
References
Chapter 4 The power of the circle
4.1 Power with respect to a circle
4.2 Golden section and regular pentagon
4.3 Radical axis, radical center
4.4 Apollonian circles
4.5 Circle pencils
4.6 Orthogonal circles and pencils
4.7 Similarity centers of two circles
4.8 Inversion
4.9 Polar and pole
4.10 Comments and exercises for the chapter
References
Chapter 5 From the classical theorems
5.1 Escribed circles and excenters
5.2 Heron’s formula
5.3 Euler’s circle
5.4 Feuerbach’s Theorem
5.5 Euler’s theorem
5.6 Tangent circles of Apollonius
5.7 Theorems of Ptolemy and Brahmagupta
5.8 Simson’s and Steiner’s lines
5.9 Miquel point, pedal triangle
5.10 Arbelos
5.11 Sangaku
5.12 Fermat’s and Fagnano’s theorems
5.13 Morley’s theorem
5.14 Signed ratio and distance
5.15 Cross ratio, harmonic pencils
5.16 Theorems of Menelaus and Ceva
5.17 The complete quadrilateral
5.18 Desargues’ theorem
5.19 Pappus’ theorem
5.20 Pascal’s and Brianchon’s theorems
5.21 Castillon’s problem, homographic relations
5.22 Malfatti’s problem
5.23 Calabi’s triangle
5.24 Comments and exercises for the chapter
References
Index