Difference between revisions of "Concurrence"

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Several [[line]]s (or [[curve]]s) are said to '''concur''' at a [[point]] if they all contain that point.
  
Several [[line]]s are said to '''concur''' at a [[point]] if they all contain that point.
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== Proving concurrence ==
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In analytical geometry, one can find the points of concurrency of any two lines by solving the system of equations of the lines.
  
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[[Ceva's Theorem]] gives a criteria for three [[cevian]]s of a triangle to be concurrent. In particular, the three [[altitude]]s, [[angle bisector]]s, [[median]]s, [[symmedian]]s, and perpendicular bisectors (which is not a cevian) of any triangle are concurrent, at the [[orthocenter]], [[incenter]], [[centroid]], [[circumcenter]], and [[Lemoine point]] respectively.
  
 
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Revision as of 18:48, 23 November 2007

This is an AoPSWiki Word of the Week for Nov 22-28

Several lines (or curves) are said to concur at a point if they all contain that point.

Proving concurrence

In analytical geometry, one can find the points of concurrency of any two lines by solving the system of equations of the lines.

Ceva's Theorem gives a criteria for three cevians of a triangle to be concurrent. In particular, the three altitudes, angle bisectors, medians, symmedians, and perpendicular bisectors (which is not a cevian) of any triangle are concurrent, at the orthocenter, incenter, centroid, circumcenter, and Lemoine point respectively.

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