At the same time we could have used the axioms set forth by Garret Birkhoff. These axioms are the ones with which you are more familiar, being the basis for the texts in High School Geometry. I have included both sets of axioms, so that you can compare the two. We will work exclusively with the Hilbert axioms. An interesting exercise is to prove everything that we will prove using Birkhoff's axioms.
Some proofs are easier, and some are harder.
First, let us set our undefined terms. These objects are to be taken as undefined terms. We will not attempt to make a definition of any of them. However, when we wish to look at a specific example of our geometry, called a model , we will have to establish how each of these undefined terms is to be interpreted. For example, if we wish to look at the model where points are ordered pairs coming from the usual Cartesian plane and lines are the usual straight lines in the Cartesian plane, then we would be able to easily state what it means for a point to be incident with a particular line, what it means for one point to lie between two other points and what it means for two line segments to be congruent.