PlanetMath: Pasch's theorem

PlanetMath

(more info)

Math for the people, by the people.

Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS

Login

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (191)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (5)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Owner confidence rating: Very high

Entry average rating: No information on entry rating

Pasch's theorem

(Theorem)

Theorem 1 (Pasch) Let $\triangle abc$ be a triangle with non-collinear vertices in a linear ordered geometry. Suppose a line $\ell$ intersects one side, say open line segment $\overline{ab}$ , at a point strictly between and , then $\ell$ also intersects exactly one of the following:

$\displaystyle \overline{bc}$ , $\displaystyle \qquad\qquad\overline{ac}$ , $\displaystyle \qquad\qquad c.$

Proof. First, note that vertices

and

are on opposite sides of line $\ell$ . Then either

lies on $\ell$ , or

does not. if

does not, then it must lie on either side (half plane) of $\ell$ . In other words,

and

must be on the opposite sides of $\ell$ , or

and

must be on the opposite sides of $\ell$ . If

and

are on the opposite sides, $\ell$ has a non-empty intersection with $\overline{ac}$ . But if

and

are on the opposite sides, then

and

are on the same side, which means that $\overline{bc}$ does not intersect $\ell$ . $\qedsymbol$

Remark A companion property states that if line $\ell$ passes through one vertex

of a triangle $\triangle abc$ and at least one other point on $\triangle abc$ , then it must intersect exactly one of the following:

$\displaystyle b$ , $\displaystyle \qquad\qquad c$ , $\displaystyle \qquad\qquad\overline{bc}.$

Of course, if $\ell$ passes through

, $\overline{ab}$ must lie on $\ell$ . Similarly, $\overline{ac}$ lies on $\ell$ if $\ell$ passes through

.

"Pasch's theorem" is owned by CWoo. [ full author list (2) ]

(view preamble)

See Also: angle, ordered geometry

Log in to rate this entry.
(view current ratings)

Cross-references: vertex, property, half plane, lie on, lies on, opposite sides, strictly, point, open line segment, side, intersects, line, linear ordered geometry, vertices, triangle
There is 1 reference to this entry.

This is version 10 of Pasch's theorem, born on 2005-10-09, modified 2007-07-27.
Object id is 7429, canonical name is PaschsTheorem.
Accessed 2032 times total.

Classification:

51G05 (Geometry :: Ordered geometries )

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

No messages.

Interact

post | correct | update request | prove | add result | add corollary | add example | add (any)