Axiom:Axiom of Segment Construction

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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 $ax$.


Sources