Cevian line: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(References: Coxeter+Greitzer)
imported>Richard Pinch
(→‎Concurrent sets: naming points of concurrency)
Line 8: Line 8:
==Concurrent sets==
==Concurrent sets==
Examples of concurrent Cevian sets include:
Examples of concurrent Cevian sets include:
* The [[altitude (geometry)|altitude]]s  
* The [[altitude (geometry)|altitude]]s, meeting at the [[orthocentre]]
* The [[median (geometry)|median]]s
* The [[median (geometry)|median]]s, meeting at the [[centroid]]
* The angle bisectors
* The angle bisectors, meeting at the [[incentre]]


==References==
==References==
* {{cite book | author=H.S.M. Coxeter | coauthors=S.L. Greitzer | title=Geometry revisited | series=New Mathematical Library | volume=19 | publisher=[[MAA]] | year=1967 | isbn=0-88385-619-0 }}
* {{cite book | author=H.S.M. Coxeter | coauthors=S.L. Greitzer | title=Geometry revisited | series=New Mathematical Library | volume=19 | publisher=[[MAA]] | year=1967 | isbn=0-88385-619-0 }}

Revision as of 15:33, 25 November 2008

In triangle geometry, a Cevian line is a line in a triangle joining a vertex of the triangle to a point on the opposite side. A Cevian set is a set of three lines lines, one for each vertex. A Cevian set is concurrent if the three lines meet in a single point.

Ceva's theorem

Let the triangle be ABC, with the Cevian lines being AX, BY and CZ. Ceva's theorem states that the Cevian set is concurrent if and only if

Concurrent sets

Examples of concurrent Cevian sets include:

References