James T. Smith


Methods of Geometry

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Wiley, 2000


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This text for the junior or senior level geometry survey course offered by many mathematics departments is distinguished by these features:

* A thorough but concise survey of elementary Euclidean geometry--in the spirit of Moïse's Elementary Geometry from an Advanced Standpoint--intended to get everyone started at the same place.
* Inclusion of classical "advanced" Euclidean geometry of triangles and circles, including most of plane and spherical trigonometry.
* Emphasis on the classification of isometries in 2 and 3 dimensions--only Martin's book is equally thorough.
* A chapter on symmetry groups, including classification of frieze and wallpaper groups comparable to Martin's, and a discussion of the structure, classification, and symmetries of polyhedra.
* A large collection of non-routine exercises, quite a few of which require considerable computation.
* Ample attention to the history and philosophy of geometry.


For the Zentralblatt review (in English), click here


Table of Contents

For more detail, select a chapter:


1 Introduction 7 Three dimensional isometries and similarities
2 Foundations of geometry 8 Symmetry
3 Elementary Euclidean geometry Appendix A: Equivalence relations
4 Exercises on elementary geometry Appendix B: Least upper bouud principle
5 Some triangle and circle geometry Appendix C: Vector and matrix algebra
6 Plane isometries and similarities Bibliography


1 Introduction


2 Foundations of geometry

2.1 Geometry as applied mathematics

2.2 Need for rigor

2.3 Axiomatic method

2.4 Euclid's Elements

2.5 Coordinate geometry

2.6 Foundation problem

2.7 Parallel axiom

2.8 Firm foundations

2.9 Geometry as pure mathematics

2.10 Exercises and projects


3 Elementary Euclidean geometry

3.1 Incidence geometry

3.2 Ruler axiom and its consequences

3.3 Pasch's axiom and the separation theorems

3.4 Angles and the protractor axioms

3.5 Congruence

3.6 Perpendicularity

3.7 The parallel axiom and related theorems

3.8 Area and Pythagoras' theorem

3.9 Similarity

3.10 Polyhedral volume

3.11 Coordinate geometry

3.12 Circles and spheres

3.13 Arcs and trigonometric functions

3.14 p


4 Exercises on elementary Euclidean geometry

4.1 Exercises on the incidence and ruler axioms

4.2 Exercises related to Pasch's axiom

4.3 Exercises on congruence and perpendicularity

4.4 Exercises involving the parallel axiom

4.5 Exercises on similarity and Pythagoras' theorem

4.6 Exercises on circles and spheres, part 1

4.7 Exercises on area

4.8 Exercises on volume

4.9 Exercises on circles and spheres, part 2

4.10 Exercises on coordinate geometry


5 Some triangle and circle geometry

5.1 Four concurrence theorems

5.2 Menelaus' theorem

5.3 Desargues' theorem

5.4 Ceva's theorem

5.5 Trigonometry

5.6 Vector products

5.7 Centroid

5.8 Orthocenter

5.9 Incenter and excenters

5.10 Euler line and Feuerbach circle

5.11 Exercises


6 Plane isometries and similarities

6.1 Transformations

6.2 Plane isometries

6.3 Reflections

6.4 Translations

6.5 Rotations

6.6 Structure theorem

6.7 Glide reflections

6.8 Isometries and orthogonal matrices

6.9 Classifying isometries

6.10 Similarities

6.11 Exercises


7 Three dimensional isometries and similarities

7.1 Isometries

7.2 Reflections

7.3 Translations and rotations

7.4 Glide and rotary reflections

7.5 Classifying isometries

7.6 Similarities

7.8 Exercises


8 Symmetry

8.1 Polygonal symmetry

8.2 Friezes

8.3 Wallpaper designs

8.4 Polyhedral symmetry

8.5 Exercises


Appendix A: Equivalence relations

Appendix B: Least Upper Bound Principle

Appendix C: Vector and matrix algebra



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