Axiom:Axiom of Segment Construction

Axiom
Let $\equiv$ be the relation of equidistance.

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

The axiom of segment construction is given by the following statement:
 * $\forall a, b, c, q: \exists x: \mathsf{B} qax \land ax \equiv bc$

where $a, b, c, q, x$ are points.

Intuition

 * AxiomOfSegmentConstruction.png

Let $bc$ be a line segment.

Let $a$ be a point on a ray with endpoint $q$.

One can create a line segment congruent to $bc$. This line segment is $ax$.