Problem 1.3#4

Theorem P1.

There exist at least four distinct lines, no three of which are concurrent.

Proof:

There exist points A, B, C, D, no three of which are collinear by Axiom P1. Lines AB, BC, CD, and AD exist by Axiom P3. We will show that these four lines are distinct and that no three are concurrent.

Suppose two of these lines are the same. Then the 3 or 4 points on the two lines would be collinear, contradicting our choice of points. Hence these four lines are distinct.

Now suppose 3 of these lines are concurrent. Without loss of generality, say AB, BC, and CD meet at P. If P = B, then B is on CD, contradicing our choice of points. If P B, then P and B lie on both of the (distinct) lines AB and BC, contradicting Axiom P3. In either case, we reach a contradiction, so that no 3 of these lines are concurrent.