Pasch's axiom
In geometry, Pasch's axiom, is a result of plane geometry used by Euclid, but yet which cannot be derived from Euclid's postulates. Its axiomatic role was discovered by Moritz Pasch.
The axiom states that, in the plane,
- A line which intersects one edge of a triangle and misses the three vertices 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.
References
- Philip J. Davis and Reuben Hersh. The Mathematical Experience. Birkhäuser Boston, Boston, 1981. Page 160. [QA8.4.D37 1982]
- Edwin Moise. Elementary Geometry from an Advanced Standpoint, Third Edition. Addison-Wesley, Reading, MA, 1990. Page 74.
External links
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