Fe-Fo Geometry

Undefined terms: Fe's, Fo's, and the relation "belongs to."

Axiom 1. There exist exactly three distinct Fe's in this system.
Axiom 2. Any two distinct Fe's belong to exactly one Fo.
Axiom 3. Not all Fe's belong to the same Fo.
Axiom 4. Any two distinct Fo's contain at least one Fe that belongs to both.

I was able to create a model for this Fe-Fo Geometry:

Axiom 1. There are exactly three people.
Axiom 2. Two distinct people belong to exactly one committee.
Axiom 3. Not all people belong to the same committee.
Axiom 4. Any two distinct committees contain one person who belongs to both.

An example of a "non-model" would be as follows...

Axiom 1. There are exactly three books.
Axiom 2. Any two books are on exactly one shelf.
Axiom 3. Not all books are on the same shelf.
Axiom 4. Any two distinct shelves contain one book that is on both.

This is a "non-model" because it is not possible for a book to be on two shelves at the same time. Axioms 2 and 4 are not "correct" statements.

We are almost to my question...

Four-Point Geometry
Axiom 1. There exist exactly four points
Axiom 2. Any two distinct points have exactly one line on both of them.
Axiom 3. Each line is on exactly two points.

My question is... What would be an example of a model and a "non-model" for Four-Point Geometry? I have attached a drawing that represents Four-Point Geometry. I am sorry for the long read.

Finite Geometry-fourpointgeo.jpg