Axiom:Inner Transitivity of Betweenness

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

This axiom asserts that:
 * $\forall a, b, c, d: \mathsf{B}abd \land \mathsf{B}bcd \implies \mathsf{B}abc$

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

Intuition

 * Inner Transitivity Axiom.png

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

Suppose they are arranged to form one straight line segment.

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

Note that 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 collinear".

Also see

 * Outer Transitivity of Betweenness