Friday, May 23, 2008

median of a triangle

In geometry, a median of a triangle is a cevian joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side.

The three medians are concurrent at a point known as the triangle's centroid, or center of mass of the triangle. Note that this means that the centroid is always in the interior of the triangle. Two-thirds of the length of each median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side.

The three medians divide the triangle into six smaller triangles of equal area.
Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.

formula for length

Applying Stewart's theorem one gets:

m = sqrt( (2 b^2 + 2 c^2 - a^2)/4 )

where a is the side of the triangle whose midpoint is the extreme point of median m.

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