Difference between revisions of "Centroid"
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<center>[[Image:centroid.PNG]]</center> | <center>[[Image:centroid.PNG]]</center> | ||
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| + | == Proof of concurrency of the medians of a triangle == | ||
| + | By [[Ceva's Theorem]], we must show that <math>AO\cdot BM\cdot CN =OB\cdot MC\cdot NA</math>. But this falls directly from the fact that <math>AO=OB, BM=MC,</math> and <math>CN=NA</math>. | ||
Revision as of 10:10, 18 August 2006
This article is a stub. Help us out by expanding it.
The centroid of a triangle is the point of intersection of the medians of the triangle and is generally denoted 
. The centroid has the special property that, for each median, the distance from a vertex to the centroid is twice that of the distance from the centroid to the midpoint of the side opposite that vertex. Also, the three medians of a triangle divide it into six regions of equal area.
The centroid is the center of mass of the triangle; in other words, if you connected a string to the centroid of a triangle and held the other end of the string, the triangle would be level.
The coordinates of the centroid of a coordinatized triangle is (a,b), where a is the arithmetic average of the x-coordinates of the vertices of the triangle and b is the arithmetic average of the y-coordinates of the triangle.
Proof of concurrency of the medians of a triangle
By Ceva's Theorem, we must show that 
. But this falls directly from the fact that 
and 
.
