Axiom:Euclid's Axiom

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

Let $=$ be the relation of equality.

This axiom asserts that:
 * $\forall a, b, c, d, t: \exists x, y:$
 * $\left({\mathsf{B}adt \land \mathsf{B}bdc \land \neg (a = d)}\right) \implies \left({\mathsf{B}abx \land \mathsf{B}acy \land \mathsf{B}xty}\right)$

where $a, b, c, d, t, x, y$ are points.

Intuition

 * Euclid'sAxiomFirstForm.png

Draw an angle $bac$ such that $0^{\circ} \le \angle{bac} \le 180^{\circ}$

Let $t$ be some point in the interior of $\angle{bac}$.

Draw a ray starting at $a$ and passing through $t$.

Let $d$ be a point on ray $at$ such that $d$ is between $a$ and $t$.

Then there is some line $xy$ passing through $t$, where $x$ is a point on one side of angle $bac$ and $y$ is a point on the other.