Question

Given the following Axioms of Fano's geometry:

1. There exists at least one line

2. Each line is on exactly 3 points

3. Not all points are on the same line

4. Each pair of points are on exactly one line

5. Each pair of lines are on at least one point

a) Prove every point is on exactly three lines

b) What geometries are possible if you eliminate Axiom 5?

Answer 1

ANS a) Pick a line l (exists by Axiom 1). Choose any point P not on l (exists by Axiom 3). Since l has 3 points (Axiom 2), joining P to each of them gives three distinct lines through P (by Axiom 4 there is a line through P and each of these points. If one of these lines contained two points of l, it would have to be l by Thm , but this contradicts the fact that P is not on l.) If there where another line through P, it would not meet l, co ntradicting Axiom 5, so there are exactly 3 lines through P. This argument takes care of all points not on l, to deal with a point on l, say Q, choose a line through P which does not contain Q and repeat the argument using Q and this line.

ANS b)  Assume, to the contrary, that distinct lines l and m, meet at points P and Q. This contradicts axiom 4, which says that the points P and Q lie on exactly one line. Thus, our assumption is false, and two distinct lines are on at most one point.