Qir nha
OK. But without your constuction, nothing distinguishes {B,C,E} from the other lines.
Your model is elegant. I like the way it illustrates the duality between points and lines that exists in projective geometry:
There are 7 points. There are 7 lines.
Each point is on 3 line. Each line contains 3 points.
[wondering][sub]Are we still in the Pit?[/sub][/wondering]
RT is correct, wrt the necessity of the point/line at infinity.
Yeah, I can see where RTFirefly’s definition of the projective plane doesn’t necessitate the idea of points or lines at infinity. However, given my definition (which we covered in my class about four weeks ago, so my memories and notes are pretty fresh) of the projective plane as a completion of the affine plane, it seems to require equivalence classes – which I termed points at infinity – to cause parallel lines in the affine plane to intersect in the projective plane.
So, is there something wrong with my definition, and if so, what?
BTW, DrMatrix, my example is the simplest/smallest projective plane, given my definition.
People ask me, Why do you love that message board so much? And I think about threads like this.
And then I think about how they would respond if I actually tried to describe these sorts of discussions. Rather than the nod of enlightenment, I visualize the other person staring blankly at me as if I had just related an experience of connubial bliss with a hedgehog formed with alternating layers of Spam and brie.
I still don’t have a good answer for the question. 
QueenAl, if the lines are in the same plane and are infinite, as MathGeek said, then I think they are parallel in the way you are thinking of. If lines (line segments) that aren’t equal distances apart along their whole length are stretched out to infinity, they will intersect. In boring plain (Euclidean?) geometry, which is all I know anything about, anyway. 
Hope that helps.
…yeah, a Pit thread about parallel lines and alternate geometries…
Nenya_Elizabeth – in Euclidean geometry, you are correct. (In fact, certain axioms dealing with the concept of “betweenness” make it necessary to have lines of infinite length) Of course, that’s not true in other geometries. In hyperbolic geometry, for example, the distance between two parallel lines increases.
For those still interested in the projective geometry issue, I think I’ve found a solution to the apparent discrepancy. RTFirefly’s third axiom states:
This is a perfectly logical axiom, and I think condenses some of mine into just one. However, if we consider the most minimal model, then this axiom only requires four points (this is the only axiom which states anything about the number of points in the projective plane). When you add the first axiom into the mix, this gives the four-point model I had mentioned, yielding three sets of parallel lines. Of course, (P2) isn’t satisfied yet. Therefore, we add points to cause those previously parallel lines to intersect. Those are the points at “infinity”. (Of course, in my model, there’s nothing “infinite” about them.) We need them to satisfy (P2).
At any rate, that’s what I think happens. It’s quite possible I have a gap in my logic somewhere. (It’s happened before…repeatedly 
)