In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, Pasch's theorem, stated in 1882 by the German mathematician
Moritz Pasch
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry. He completed his Ph.D. at the Univer ...
,
is a result in
plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
which cannot be derived from
Euclid's postulates.
Statement
The statement is as follows:
ere, for example, (, , ) means that point lies between points and .
See also
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Ordered geometry Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affi ...
*
Pasch's axiom
Notes
References
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*
External links
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Euclidean plane geometry
Foundations of geometry
Order theory
Theorems in plane geometry
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