Contents

Preface

Acknowledgments

1 Introduction

1.1 A Short History of Geometry

1.2 What you will and will not learn in this book

1.3 Audience prerequisites and style of explanation

1.4 Book plan

1.5 How to study this book

2 Preliminaries

2.1 Proof methods

2.1.1 Methods for proving conditional statements

2.1.2 Methods for proving other types of statements

2.1.3 Symbolic representation

2.1.4 More examples of proofs

2.1.5 Exercises

2.2 Elementary theory of sets

2.2.1 Set operations

2.2.2 Relations

2.2.3 Equivalence relations

3 Incidence geometry

3.1 The notion of incidence geometry

3.2 Lines and collinearity

3.3 Examples of incidence geometries

3.3.1 Some basic examples of incidence geometries

3.3.2 The main incidence geometries

3.3.3 Generalizing the real cartesian plane

3.4 Parallelism

3.5 Behavior of parallelism in our examples

4 Betweenness

4.1 Betweenness structures, segments, triangles, and convexity

4.2 Separation of the plane by a line

4.3 Separation of a line by one of its points

4.4 Rays

4.5 Angles

4.6 Betweenness structure for the real cartesian plane

4.7 Betweenness structure for the hyperbolic plane

5 Congruence of segments

5.1 Congruence of segments structure and segment comparison

5.2 The usual congruence of segments structure for the real cartesian plane

5.3 The usual congruence of segments structure for the hyperbolic plane

6 Congruence of angles

6.1 Congruence of angles structure and angle comparison

6.2 Angle congruence in our main examples

6.2.1 Congruence of angles in the real cartesian plane

6.2.2 Congruence of angles in the hyperbolic plane

7 Hilbert planes

7.1 Circ1es

7.2 Book 1 of The Elements

References

Index

Filtros

a Course of Plane Geometry

A course of plane geometry