In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, Pasch's axiom is a result of plane geometry

Plane geometry

In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...

 used by Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

, but yet which cannot be derived from Euclid's postulates. Its axiomatic role was discovered by Moritz Pasch

Moritz Pasch

Moritz Pasch was a German mathematician specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age...

.

The axiom states that, in the plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

,

A line

Line (mathematics)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

 which intersects

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

 one edge of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 and misses the three vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 must intersect one of the other two edges.

Pasch published this axiom in 1882, and showed that Euclid's axioms were incomplete.

In other treatments of elementary geometry, Pasch's axiom can be proved as a theorem; it is a consequence of the plane separation postulate.

Pasch's axiom is distinct from Pasch's theorem

Pasch's theorem

In geometry, Pasch's theorem, stated in 1882 by a German mathematician Moritz Pasch, is a result of plane geometry which cannot be derived from Euclid's postulates. It would now be considered as order theory, but the point it makes is in relation to the axiomatic method.The statement is as follows...

.