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

Axiom
This is Pasch's Axiom in the language of Tarski's Geometry.

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 $ac$, $bc$, and $qa$ be line segments.

Let line segments $ac$, $bc$, and $qa$ form a triangle.

Let $cb$ be a ray with endpoint $c$.

Let $q$, which is a vertex of $\triangle{acq}$, be on ray $cb$.

Let $p$ be a point on the line segment $ac$.

Then:

Draw a line segment connecting points $p$ and $b$.

Then line segment $pxb$ will intersect segment $qxa$ 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 $ac$, $qb$, and $aq$ be line segments.

Also see

 * First and Second Forms of Tarski's Pasch Axiom are Equivalent