Axiom:Pasch's Axiom (Tarski's Axioms)

Axiom
Let $a,b,c,p,q,x$ be points.

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

First form
The first form of the axiom is:


 * $\forall a,b,c,p,q \ \exists x :\mathsf{B}apc \land \mathsf{B}bqc \implies \mathsf{B}pxb \land \mathsf{B}qxa$

Intuition

 * Tarski's Inner Pasch Axiom.png

Let $aqc$ be a triangle.

Draw a line segment extending segment $cq$ to some point $b$ outside the triangle such that $c, q, b$ are collinear.

Pick a point $p$ on segment $ac$.

Draw a line segment connecting point $p$ with point $b$.

Segment $pb$ will intersect segment $aq$ at some point $x$.

Second form
The second form of the axiom is:


 * $\forall a,b,c,p,q \ \exists x : \mathsf{B}apc \land \mathsf{B}qcb \implies \mathsf{B}axq \land \mathsf{B}bpx$

Intuition

 * Tarski's Outer Pasch Axiom.png

Let $a, p, c$ be collinear.

Further, let $q,c,b$ be collinear.

Construct a ray with endpoint $a$ passing through $q$.

Construct another ray with endpoint $b$ passing through $p$.

Ray $aq$ and ray $bp$ will intersect at some point $x$.

Also see

 * Equivalence of Formulations of Pasch's Axiom