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

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This page is about Pasch's Axiom in Tarski's Geometry. For other uses, see Axiom:Pasch's Axiom.

Axiom

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


First form

The first form of the axiom is:

$\forall a, b, c, p, q: \exists x :\mathsf B a p c \land \mathsf B b q c \implies \mathsf B p x b \land \mathsf B q x a$

where $a, b, c, p, q, x$ are points.


Intuition


Tarski's Inner Pasch Axiom.png

Let $a q c$ be a triangle.

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

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

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

Segment $p b$ will intersect segment $a q$ at some point $x$.


Second form

The second form of the axiom is:

$\forall a, b, c, p, q: \exists x : \mathsf B a p c \land \mathsf B q c b \implies \mathsf B a x q \land \mathsf B b p x$

where $a, b, c, p, q, x$ are points.


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


Source of Name

This entry was named for Moritz Pasch.


Sources