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# Counting Points and Lines Using Axioms

Asked by Bob Williams on January 9, 1998:
Suppose I am given that
• If P and Q are two points, there is exactly one line containing P and Q
• If L is any line, there is a point P which does not lie on L
• There are at least three points on every line
• Any two distinct lines intersect at exactly one point
• There exists at least one line.
How do I prove that, if one line contains exactly n points, then
1. Every line contains exactly n points?
2. Every point lies on exactly n lines?
3. The space contains n^2 - n +1 points and n^2 - n + 1 lines?
The following hints should help you answer this question.

First, try proving that, for any two lines L and M, there is at least one point R not on either of them. Do this using the fact that L and M each have at least three points, so you can find a point P on L which isn't the intersection point, and a point Q on M which isn't the intersection point. The line PQ will have a third point R on it (because every line has at least three points). R is not on L (if it were, L and PQ would both be lines containing R and P. Since there is a unique line joining any two points, L would have to equal PQ, contradicting the fact that Q is not on L). Similarly, R is not on M.

Now you can prove that any two lines L and M have the same number of points. They have their intersection point X in common. For each remaining point P on L, the line RP intersects M in a point f(P). Show that f establishes a 1-1 correspondence between the points on L (other than X) and the points on M (other than X).

To prove part (2), for any point P, show that there must be at least one line L not containing P. Every line through P intersects L in a point. Show that this process sets up a 1-1 correspondence between lines through P and points on L, proving that P is on exactly n lines.

Now pick a point P. Every other point must lie on a line through P. There are n such lines, with n-1 points on each, so there are n(n-1) points not including P. Thus there are n(n-1)+1 points in total. A similar argument gives you the number of lines.

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