Axiom:Inner Connectivity of Betweenness

Axiom
Let $a,b,c,d$ be points.

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

This axiom asserts that:


 * $\forall a,b,c,d: \left({\mathsf{B}abd \land \mathsf{B}acd}\right) \implies \left({\mathsf{B}abc \lor \mathsf{B}acb}\right)$

Intuition
Let $abd$ and $acd$ be line segments.

They exist in one of the following configurations:

Case 1


Point $b$ is between $a$ and $c$.

Case 2


Point $c$ is between $a$ and $b$.

Note that this axiom does not assert that exactly one of the cases happened.

This axiom still holds in the degenerate cases where the points are not distinct.

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

Also see

 * Outer Connectivity of Betweenness