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).