So the axioms for a projective plane are given by:
- Any two “points” are contained in a unique “line.”
- Any two “lines” contain a unique “point.”
- There exist four “points”, no three of which are in a “line.”
Consider now this question: Suppose that $A,B,C,D$ are four “points” in a projective plane, no three of which are in a “line.” Consider the “lines” $AB,BC,CD,DA$. Show that if $AB$ and $BC$ have a common point $E$, then $E = B$.
I feel like I'm not approaching this correctly. To me it just seems that if $E\neq B$, then $AB$ and $BC$ intersect at $B$, and also intersect at $E$, which contradicts axiom 2. But this does not use the hypothesis at all. I think I'm missing something very, very obvious.