# Geometry/Neutral Geometry/Axioms of Betweenness

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# Axioms of Betweenness[edit | edit source]

## Betweenness Axiom 1[edit | edit source]

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

### Explanation[edit | edit source]

## Betweenness Axiom 2[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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.

### Explanation[edit | edit source]

## Betweenness Axiom 4[edit | edit source]

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[edit | edit source]

- (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*.