Cevian line: Difference between revisions
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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. | 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. | ||
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==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 }} | ||
Latest revision as of 22:12, 7 February 2009
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:
- The altitudes, meeting at the orthocentre
- The medians, meeting at the centroid
- The angle bisectors, meeting at the incentre
References
- H.S.M. Coxeter; S.L. Greitzer (1967). Geometry revisited. MAA. ISBN 0-88385-619-0.