Two Coordinatization Theorems for Projective Planes

Since the points of a line uniquely determine the line, it is safe to talk about lines as sets of points, but for convenience we will often regard them separately. We will talk about projective planes using the usual language of points and lines “lies on”, “passes through”, “collinear”, “concurrent”, etc. The third requirement implies its dual, that there are four lines, no three of which are concurrent (consider all the lines determined by the four points), and vice versa, so for any plane we can form the dual plane by swapping the points and lines. Thus for any statement true for all projective planes, the dual statement is also true. A system satisfying the first two requirements but not the third is called a degenerate projective plane. We will not prove this here, but by case analysis it is can be seen that all degenerate projective planes are as follows[1]: