Pasch's theorem

Not to be confused with Pasch's axiom regarding a line through a triangle

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch,^[1] is a result in plane geometry which cannot be derived from Euclid's postulates.

Statement

The statement is as follows:

Pasch's theorem — Given points $a$ , $b$ , $c$ , and $d$ on a line, if it is known that the points are ordered as ( $a$ , $b$ , $c$ ) and ( $b$ , $c$ , $d$ ), then it is also true that ( $a$ , $b$ , $d$ ).^[2]

[Here, for example, ( $a$ , $b$ , $c$ ) means that point $b$ lies between points $a$ and $c$ .]

Notes

^ Pasch 1912
^ Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.

References

Coxeter, H.S.M. (1969), Introduction to geometry (2nd ed.), John Wiley and Sons, ISBN 978-0-471-18283-2, Zbl 0181.48101
Pasch, Moritz (1912) [first edition 1882], Vorlesungen uber neuere Geometrie (in German) (2nd ed.), Leipzig: B.G. Teubner

External links

Weisstein, Eric W. "Pasch's Theorem". MathWorld.

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[1] Pasch 1912

[2] Coxeter (1969, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem.

[1]

[2]

Statement

Notes

References

See also

External links