Axioms of Betweenness

Betweenness Axiom 1

If A*B*C, then A,B, and C are three distinct points all lying on the same line, and C*B*A.

Betweenness Axiom 2

Given any two distinct points B and D, there exist points A,C, and E lying on line BD (needs format with LaTex) such that A*B*D, B*C*D, and B*D*E.

Explanation

suppose that in a certain metric gemetry the following distance relationship hold: AB= 2 AD=BD=CD=3 BC=4 AC=6

Betweenness Axiom 3

If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.

Betweenness Axiom 4

For every line l and for any three points A, B, and C not lying on l:

• (i) If A and B are on the same side of l and if B and C are on the same side of l, then A and C are on the same side of l.
• (ii) If A and B are on opposite sides of l and if B and C are on opposite sides of l, then A and C are on the same side of l.

Corollary

• (iii) If A and B are on opposite sides of l and if B and C are on the same side of l, then A and C are on opposite sides of l.