Axiom:Pasch's Axiom (Tarski's Axioms)
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This page is about Pasch's Axiom in the context of Tarski's Geometry. For other uses, see 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
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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
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 $x$.
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Also see
Source of Name
This entry was named for Moritz Pasch.
Sources
- June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $179, 180$ : Axiom $7$
Illustration courtesy of Steven Givant.