Another Geometric Structure
We will consider a structure for creating a configuration of triangles
and their vertices.
Conditions for a Geometric Structure

Any two triangles will exactly one point in common.

Any two points determine a unique triangle.

There are at least two triangles.
Here are two examples:
Notice that each structure has 7 triangles and 7 vertices!
C 
E 
G 


G 
B 
C 
B 
D 
G 


G 
D 
F 
A 
F 
G 


G 
A 
E 
A 
D 
E 


C 
E 
D 
B 
F 
E 


D 
A 
B 
A 
B 
C 


C 
A 
F 
D 
F 
C 


B 
F 
E 
Interesting... the question we is still:
Are there any of these geometric
stuctures that have more or less than 7 triangles made from 7 vertices?
That is, is there some special
relation between these geometric structures and the number 7?