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Geometry Revisited
H.S.M. Coxeter; S.L. Greitzer
ISBN
9789393330161
Language
English
Pages
208
Format
Paperback
Dimensions
158 x 240 mm
Year of Publishing
2022
Series
Catalogues

Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morleys remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed.

H. S. M. Coxeter, S. L. Greitzer

Preface

Chapter 1 Points and Lines Connected with a Triangle

1.1 The extended Law of Sines
1.2 Ceva´s theorem
1.3 Points of interest
1.4 The incircle and excircles
1.5 The Steiner-Lehmus theorem
1.6 The orthic triangle
1.7 The medial triangle and Euler line
1.8 The nine-point circle
1.9 Pedal triangles

Chapter 2 Some Properties of Circles
2.1 The power of a point with respect to a circle
2.2 The radical axis of two circles
2.3 Coaxal circles
2.4 More on the altitudes and orthocenter of a triangle
2.5 Simson lines
2.6 Ptolemy´s theorem and its extension
2.7 More on Simson lines
2.8 The Butterfly
2.9 Morley´s theorem

Chapter 3 Collinearity and Concurrence
3.1 Quadrangles; Varignon´s theorem
3.2 Cyclic quadrangles; Brahmagupta´s formula
3.3 Napoleon triangles
3.4 Menelaus´s theorem
3.5 Pappus´s theorem
3.6 Perspective triangles; Desargues´s theorem
3.7 Hexagons
3.8 Pascal´s theorem
3.9 Brianchon´s theorem

Chapter 4 Transformations
4.1 Translation
4.2 Rotation
4.3 Half-tum
4.4 Reflection
4.5 Fagnano´s problem
4.6 The three jug problem
4.7 Dilatation
4.8 Spiral similarity
4.9 A genealogy of transformations

Chapter 5 An Introduction to Inversive Geometry
5.1 Separation
5.2 Cross ratio
5.3 Inversion
5.4 The inversive plane
5.5 Orthogonality
5.6 Feuerbach´s theorem
5.7 Coaxal circles
5.8 Inversive distance
5.9 Hyperbolic functions

Chapter 6 An Introduction to Projective Geometry
6.1 Reciprocation
6.2 The polar circle of a triangle
6.3 Conics 138 6.4 Focus and directrix
6.5 The projective plane
6.6 Central conics
6.7 Stereographic and gnomonic projection
Hints and Answers to Exercises
References
Glossary
Index

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