Coordinatization

In this first section, we will assign coordinates to the points of an arbitrary affine plane in the following manner. Let be an affine plane. Choose any point, and name it 0. Using the proposition that an affine plane has four points, and three are noncollinear, we know that there exists three lines through our point 0 which are not parallel (by axiom A1). Let these three lines be named the x-axis, the y-axis, and the diagonal. Choose a point on the diagonal different from 0 and name it I. Let be any abstract set with the same cardinality as the number of points on the diagonal. Let two elements of be 0 and 1. Now choose and fix a bijection
(points on diagonal) such that and . We will use the elements of to be the coordinates of the points of in the following manner.

Diagonal points

If a point A is on the diagonal, and , let (a,a) be it's coordinate.

Non-diagonal points

Suppose P is a point not on the diagonal. By axiom A2, there exists a line through P parallel to the x-axis. This line must intersect the diagonal, say at a point B with coordinates (b,b). Similarily, there exists a line through P and parallel to the y-axis which must intersect the diagonal. Let this point be C with coordinates (c,c). Assign the coordinate (c,b) to point P.

 
Figure 1: The point P(c,b)

We can also describe the lines of the plane by defining slope and y-intercept. By A2 there exists the line through the point I parallel to the y-axis. Let this line be the line of slopes. Given any line l not parallel to the y-axis, we know there exists a line parallel to it and through the point 0. This line intersects the line of slopes at a point (1,m). Assign the slope m to l. The point at which l intersects the y-axis is (0,b), and we call the y-intercept b. Note that the slope of the diagonal is 1, the slope of the x-axis is 0, and we take the slope of the y-axis and all other lines in its pencil to have undefined slope. The slope and y-intercept uniquely determine the lines of .

 
Figure 2: Slope and y-intercept

Given this coordinatization, we can define a ternary operation,
by the following. Let l be a line with slope m and y-intercept b. Let s be a line through (a,0) parallel to the y-axis. Define the point of intersection of l and s to be (a,T(a,m,b)).

 
Figure: Definition of

Notice that a point P with coordinates (x,y) lies on the line l if and only if the equation is satisfied. Thus all lines have equations of the form or x=a, a a constant for lines parallel to the y-axis.