Axiom:Outer Transitivity of Betweenness

Axiom

Let $\mathsf{B}$ be the relation of betweenness.

Let $=$ be the relation of equality.

This axiom asserts that:

$\forall a, b, c, d: \mathsf{B}abc \land \mathsf{B}bcd \land \neg \paren {b = c} \implies \mathsf{B}abd$

where $a, b, c, d$ are points.

Intuition

Let $abc$ and $bcd$ be line segments.

Suppose they are arranged to form one straight line segment.

Then the points $a, b, d$ are collinear.

Note that this axiom still holds in the degenerate cases where not all the points are not (pairwise) distinct.

For example, if we are dealing with exactly three points, this axiom could be interpreted as "three points on a line are collinear".

Also see

• Inner Transitivity of Betweenness
• Pasch's Theorem

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $186$ : Axiom $16$