Proof: Betweenness: Proposition 1
Between: Proposition 1
Using the Axioms of Independence and betweenness and the line separation property, show that sets of four points A, B, C, D on a line behave as we expect them to with respect to betweenness. Namely, show that:
1) A*B*C and B*C*D imply A*B*D and A*C*D
2) A*B*D and B*C*D imply A*B*C and A*C*D
1) Consider a line, , with points A, B, C, D on the line in the following order: A*B*C and B*C*D. By Axiom B1, we know that B lies between points A and C, while C lies between B and D. Note that B separates the line segment AD into two sets: one set containing A, and one set containing C and D. Therefore, because of the order of our points, we can conclude that A*B*D. Similarly, we can use C to separate segment AD where one set contains points A and B, and the other set contains D. Thus, we have A*C*D.
2) Consider a line, , with points A, B, C, D on the line in the following order: A*B*D and B*C*D. By Axiom B2, we know that B lies between points A and D, while C lies between B and D. Note that B separates the line segment AD into two sets, one set contain A, and one set containing C and D. Similarly to part 1, we can conclude that A*B*C. Because we can also separate segment AD into two sets using point C, we can then use similar logic to determine that A*C*D.