Math 3210 Quiz # 1 Fall 2006

Name ______________________________

In the 4-line geometry, whose axioms are:

1. There exist exactly four lines.

2. Any two distinct lines have exactly one point on both of them.

3. Each point is on exactly two lines.

Prove that not all points lie on a single line.

Solution:
BWOC assume that all the points of the geometry lie on a line m. There are (at least) two other lines, l and n, by Axiom 1. Each of these meets m by Axiom 2. The points of intersection are distinct since otherwise there would be three lines on a point contradicting Axiom 3. By Axiom 2, l and n must have a common point and this point can not be on m, contradicting our assumption.