Axiom:Euclid's Axiom

First Form
Let $a,b,c,d,t,x,y$ be points.

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

Let $\ne$ be the relation of being distinct.

This axiom asserts that:

$\forall a,b,c,d,t : \exists x,y:$


 * $\left({\mathsf{B}adt \land \mathsf{B}bdc \land a \ne d}\right) \implies \left({\mathsf{B}abx \land \mathsf{B}acy \land \mathsf{B}xty}\right)$

Intuition


Draw an angle $bac$.

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

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

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.

Also see

 * Euclid's Fifth Postulate