Thread: Three point geometry-easy?

Axioms for the Three Point Geometry:
1. There exist exactly 3 points in this geometry.
2. Two distinct points are on exactly one line.
3. Not all the points of the geometry are on the same line.
4. Two distinct lines are on at least one point.

Can someone come up with a shape that satisfies axioms 1-3 but not 4? Trying to show that the axioms are independent and I've gotten every one except axiom 4.

Thx!

Originally Posted by sfspitfire23

Axioms for the Three Point Geometry:
1. There exist exactly 3 points in this geometry.

Call them "A", "B", and "C".

2. Two distinct points are on exactly one line.
3. Not all the points of the geometry are on the same line.

There must exist lines that contain A and B but not C; A and C but not B; and B and C but not A: call them {AB}, {AC}, and {BC}

4. Two distinct lines are on at least one point.

Add a fourth line, {A} that lies on only point. The geometry consisting of points A, B, and C and lines {AB}, {BC}, {AC}, and {A}, satifies axoims 1, 2, and 3, but not 4.

[/quote]Can someone come up with a shape that satisfies axioms 1-3 but not 4? Trying to show that the axioms are independent and I've gotten every one except axiom 4.

Thx![/QUOTE]