Always a point between two others

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SUMMARY

The discussion focuses on proving that for any distinct points A and B, there exists a point X between them, utilizing the Pasch axiom from Hilbert's axioms for plane geometry. The user seeks assistance in justifying specific steps related to points P, Q, and R, particularly steps 6-8, which involve demonstrating that certain points cannot lie on the line segment defined by R and P. The need for clear logical reasoning based on established axioms is emphasized, particularly in visualizing the relationships between the points.

PREREQUISITES
  • Understanding of Hilbert's axioms for plane geometry
  • Familiarity with the Pasch axiom
  • Basic knowledge of geometric constructions
  • Ability to visualize geometric relationships
NEXT STEPS
  • Study the implications of the Pasch axiom in geometric proofs
  • Explore Hilbert's axioms in detail, focusing on their applications
  • Practice geometric constructions to reinforce understanding of point relationships
  • Review examples of constructive proofs in geometry
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Students of geometry, educators teaching geometric proofs, and anyone interested in the foundations of plane geometry and logical reasoning in mathematics.

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

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