Always a point between two others

[PingPong]

Homework Statement

Below is a sketch for a proof that for any distinct points A and B, there is always a point X between them:

Take P not on $$\overleftrightarrow{AB}$$ and Q with P between B and Q. Now take R with Q between A and R. The Pasch axiom shows that $$\overleftrightarrow{RP}$$ crosses AB.

Write down justifications for the steps below:
1) Why is there P not in $$\overleftrightarrow{AB}$$?
2) Why is there Q with P between B and Q?
3) Why is Q not equal to A?
4) Why is there R with Q between A and R?
5) Why is R not equal to P?
6) Why is B not on $$\overleftrightarrow{RP}$$?
7) Why is A not on $$\overleftrightarrow{RP}$$?
8) Why is Q not on $$\overleftrightarrow{RP}$$?
9) Why does $$\overleftrightarrow{RP}$$ not cross AQ?

Homework Equations

The Hilbert axioms for plane geometry.

The Attempt at a Solution

I've been able to get the first 5 steps which are quite easy. But 6-8 have been giving me trouble. I can see that they're important because they're setting up using the Pasch axiom, but I can't figure out why they are true based by starting with the axioms.

For 6, I can see (if I draw a picture) then R=Q, which means that R is between A and itself (which can't be). So I can intuitively see that they don't work, but I'm not sure how to put it together.

Any help please? Thanks in advance.

 

[Defennder]

Er, couldn't you just find do a quick constructive proof by finding the midpoints between two points?