Difference between revisions of "Concurrence"

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{{WotWAnnounce|week=Nov 22-28}}
 
{{WotWAnnounce|week=Nov 22-28}}
  
Several [[line]]s (or [[curve]]s) are said to '''concur''' at a [[point]] if they all contain that point.
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Several (that is, three or more) [[line]]s (or [[curve]]s) are said to be '''concurrent''' at a [[point]] if they all contain that point. The point is said to be the point of concurrence.
  
 
== Proving concurrence ==
 
== 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.
 
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 [[cevian]]s of a triangle to be concurrent. In particular, the three [[altitude]]s, [[angle bisector]]s, [[median]]s, [[symmedian]]s, and [[perpendicular bisector]]s (which are not cevians) of any triangle are concurrent, at the [[orthocenter]], [[incenter]], [[centroid]], [[Lemoine point]], and  [[circumcenter]], respectively.  
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==Related Theorems==
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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 bisector]]s (which are not cevians) of any triangle are concurrent, at the [[orthocenter]], [[incenter]], [[centroid]], [[Lemoine point]], and  [[circumcenter]], respectively.  
  
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==Problems==
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===Introductory===
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*Are the lines <math>y=2x+2</math>, <math>y=3x+1</math>, and <math>y=5x-1</math> concurrent? If so, find the the point of concurrency. ([[Concurrence/Problems#Introductory|Source]])
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===Intermediate===
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===Olympiad===
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[[Category:Definition]]
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[[Category:Geometry]]
 
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Revision as of 20:29, 23 November 2007

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

Several (that is, three or more) lines (or curves) are said to be concurrent at a point if they all contain that point. The point is said to be the point of concurrence.

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.

Related Theorems

Problems

Introductory

  • Are the lines $y=2x+2$, $y=3x+1$, and $y=5x-1$ concurrent? If so, find the the point of concurrency. (Source)

Intermediate

Olympiad

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