Three Point Geometry
Consider the following system of axioms concerning the undefined terms point and line:
1.
2.
3.
4.
There exist exactly three distinct points.
Each two distinct points lie on exactly one line.
Each two distinct lines intersect in at least one point.
Not all the points are on the same line.
QUESTIONS:
A. Provide a model of this geometry.
B. Prove that each two distinct lines intersect in exactly one point.
(Hint: Suppose two distinct lines intersected in more than one point. What does it mean
to be distinct?)
C. Prove that there are exactly three lines in the geometry.
(Hint: You must prove both that there are at least 3 and no more than 3 lines. You might
want to give the three points names, and come up with ways to name lines.)
Four Line Geometry
Axioms:
1.
2.
3.
There exist exactly four lines.
Each pair of lines has exactly one point in common.
Each point is on exactly two lines.
Prove:
1.
2.
The four line geometry has exactly six points.
Each line of the four line geometry has exactly three points on it.
Questions:
A..
B.
Does each pair of points in the geometry lie on exactly one line? Justify.
Does the geometry have any parallel lines (lines with no point in common)?