﻿ Pasch's axiom

# Pasch's axiom

The set of Pasch (after Moritz Pasch ) is commonly used in synthetic geometry as an axiom:

" Let A, B, C three not located in a straight line and a point is a straight line in the plane ABC, which meets any of these three points. Then, if the line goes through a point of a line segment AB, it is certainly also by either a point of the segment BC or a point of the segment AC. "

Clearly, this can be expressed as: "If a straight line enters through one side to the interior of a triangle, it certainly occurs again by a side of the triangle out. "

Pasch has this axiom formulated in 1882. Euclid was not interested nor for the need for such an axiom. Evidence of this kind were used by him (and his disciples in the following 2000 years ) quite naturally.

Therefore, the formulation of this axiom represents an important step on the road to geometry to a strictly axiomatic theory ( axiomatization ). It is one of the axioms can be characterized by a weak inter-relationship on an affine plane. In Hilbert's system of axioms of Euclidean geometry, it is one of the axioms that describe a (strong ) inter-relationship and thus an arrangement of the plane.

→ Also, the Axiom of Veblen -Young has been referred to in the mathematical literature as an axiom of Pasch.

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