How to study this book
2.1.1 Methods for proving conditional statements
2.1.2 Methods for proving other types of statements
3.1 The notion of incidence geometry
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
3.5 Behavior of parallelism in our examples
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
4.6 Betweenness structure for the real cartesian plane
4.7 Betweenness structure for the hyperbolic plane
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
6.1 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
1.1 A Short History of Geometry
It is safe to say that the first geometric facts recorded in human history are found within the Egyptian and the Babylonian civilizations. There is strong evidence suggesting that even the Pythagorean Theorem was well known to these civilizations. However, these discoveries were only empirical facts, geometrical regularities that seemed to occur in every case considered. From this evidence, they would come to believe that these were universally true statements, although it seems that nobody bothered to find out why these phenomena took place, or
