A Brief History of Geometry
Thales (640-546 B.C.)
abstraction of points and lines

Euclid (about 300 B.C.)
writes the "Halliday and Resnick" of geometry texts

Postulates (as worded by Ryan's teacher H. S. M. Coxeter):
I. A straight line may be drawn from any point to any other point.
II. A finite straight line may be produced to any length in a straight line
III. A circle may be described with any centre at any distance from that centre.
IV. All right angles are equal.
V. If a straight line meet two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.

Archimedes of Syracuse (287-212 B.C.) approximates the area of a circle using inscribed and circumscribed polygons.

Pappus of Alexandria (4th century A.D.) states and proves theorems of projective geometry.

Descartes (1596-1650) invents analytic geometry.

1700s: Many mathematicians make futile attempts to prove Euclid's Postulate V from the other four, but only end up by discovering alternative versions of the postulate.

Early 1800s: Gauss, Bolyai, and Lobachevsky independently discover hyperbolic geometry, which satisfies Euclid's first four postulates but not the fifth.

Klein(1849-1925) and others invent models within Euclidean geometry that satisfy the axioms of hyperbolic geometry.

In 1872 Klein's "Erlanger Programm" redefines geometry as the study of properties of figures that are invariant under a group of motions.

Riemann (1826-1866) invents new geometries in which the "lines" are geodesics on surfaces, thereby bringing calculus into geometry.

Lie(1842-1899) organizes the study of the types of groups of motions that are relevant to geometry.

Hilbert(1862-1943) redoes the axiomatic basis of geometry.

Einstein(1879 - 1955) formulates the theory of relativity, which provides a physical model for some non-Euclidean geometries.

Sommerville(in 1910) classifies the nine Cayley-Klein plane geometries.

Approaches to geometry
Axiomatic : state axioms about undefinable entities like points and lines, and prove theorems that follow logically from them.

Axioms must be consistent, but this cannot be proved within the axiomatic system.  The solution: invent models in which a set of objects whose existence is verifiable satisfies all the axioms.
The objects may be
1. Things in the physical world, which can be studied experimentally.
2. Real numbers, which are widely believed to exist and to obey the axioms of arithmetic.
3. Euclidean points, lines, circles, etc. satisfy Euclid's or Hilbert's (presumably consistent) axioms.
4. "Objects" of a programming language like C++ or Java.  Postulates or theorems about constructions are implemented as functions; other theorems become ASSERT statements. 

Other properties of sets of axioms:

"Independent" means that no axiom can be deduced from the others.  This can be established by constructing a model in which the negation of one of the axioms holds.
"Categorical" means that there are sufficient axioms to single out one geometry rather than a family of geometries.

Analytic: Study models of geometry based on real numbers (objects are vectors, motions are represented by matrices).  This is the approach that is closest to today's mathematics curriculum.  It lends itself well to computer implementation.

Empirical: Study geometries for which the physical world provides models.  (Then it becomes important to determine experimentally whether the angles of a triangle sum to exactly p/2.)

Plane affine geometry is a form of incidence geometry that studies the consequences of Euclid's first and fifth postulates, in the form

A1. Two points determine a unique line.
A2. Given a line and a point not on that line, there is precisely one line through that point that does not intersect the given line (called parallel to it.)

Statements in affine geometry may involve concepts like "parallel," "intersect," "collinear," "parallelogram," but not words like "congruent," "distance," "perpendicular," or "rectangle."  Most theorems of high-school geometry are not theorems of affine geometry.

Here is a famous statement of affine geometry, generally known as the "converse of Desargues' theorem."
Let A, B, C, A', B', C' be six distinct points (defining two triangles).
AB is parallel to A'B', AC is parallel to A'C', and BC is also parallel to B'C'.
Then the lines AA', BB', CC' either are concurrent or are all parallel.  

What is the status of this statement?

Draw a diagram on paper to illustrate it in Euclidean geometry, and it appears true.
Use high-school techniques involving similar triangles, and it is a provable theorem of Euclidean geometry.
Define the affine plane to be R2 (Math 21a) or introduce R2 as an analytic model for the affine plane (Ryan) and it is easy to prove the statement using vectors. (proof to be done on blackboard)
Assume as an additional axiom the existence of more than one affine plane, where two affine planes intersect in a line, and the statement is provable from the axioms.

The Moulton Plane (1902)

Moulton points are the same as Euclidean points.
The dotted line in the diagram below is the x-axis.
Any Euclidean line of negative slope, or parallel to the x or y axis, is a Moulton line. A broken Euclidean line of positive slope, whose slope below the x axis is twice as great as its slope above the x axis, is also a Moulton line.

Moulton lines and Moulton points satisfy the axioms above.  Two points, even one above the x-axis and one below it, determine a unique line.  To construct the unique line parallel to a given line through a given point, use the Euclidean construction above or below the x-axis, then extend the line through the x-axis.

But the diagram below shows a counterexample to the Desargues statement!!!!