Definitions
Most geometry books start off with points, lines and planes. They usually wave their hands and give some vague excuses for not defining these terms when perfectly good definitions exist. It would be more accurate to say "the definitions are very long and complicated and we aren't going to bother you with them since you probably have a pretty good idea what they are already anyway."^{1} With that said let us make some points about them:
The segment from A to B is made up of the points A and B and all the points between^{2} them. The segment between A and B is written . The ray from A through B consists of and all points P such that B is between A and P. That's pretty confusing. Look at it this way: A ray is like a segment but it keeps going in one direction (where a line keeps going in both directions). A is called the endpoint of the ray. The ray from A through B is written . The figure to the right has examples of all of these objects. First, let's take note of a drawing convention. The dotted part of is behind plane m. The implication is that crosses the plane but isn't contained in the plane. Figure 1 has one plane, m, and seven points, A through G. There are two lines: and . Rays are a little more complicated. First there is which contains . Both of these can be reversed to get and . Finally we have . Because there is nothing drawn to the left of B there is no ray from C through B. Finally, the figure contains four segments: , , and . ^{1} If you have some time on your hands and are really bored you can look them up in David Hilbert's Foundations of Geometry. ^{2} Between has a technical definition: A point C is between A and B if |