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Always a point between two others

  1. Sep 11, 2008 #1
    1. The problem statement, all variables and given/known data
    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 [tex]\overleftrightarrow{AB}[/tex] and Q with P between B and Q. Now take R with Q between A and R. The Pasch axiom shows that [tex]\overleftrightarrow{RP}[/tex] crosses AB.

    Write down justifications for the steps below:
    1) Why is there P not in [tex]\overleftrightarrow{AB}[/tex]?
    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 [tex]\overleftrightarrow{RP}[/tex]?
    7) Why is A not on [tex]\overleftrightarrow{RP}[/tex]?
    8) Why is Q not on [tex]\overleftrightarrow{RP}[/tex]?
    9) Why does [tex]\overleftrightarrow{RP}[/tex] not cross AQ?

    2. Relevant equations

    The Hilbert axioms for plane geometry.

    3. 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.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 11, 2008 #2

    Defennder

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    Homework Helper

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