Four-line geometry axioms:


Axiom 1.  There exist exactly four lines.
Axiom 2.  Any two distinct lines have exactly one point on both of them.
Axiom 3. Each point is on exactly two lines.

Draw a model for the geometry and prove the following theorems:

Theorem 1: The geometry has exactly six points.
Theorem 2: Each line had exactly three points on it.
Theorem 3: Parallel lines do not exist.

By exchanging the words "line" and "point" in the axioms (and making other necessary language changes), we can formulate plane duals of the axioms and create a four-point geometry.

Formulate duals for axioms above.
Draw a model for the new geometry.
Formulate duals for theorems and prove them (without using duality).