TOPICS
Triangle Median
A triangle median 
of a triangle 
is the Cevian from one of its vertices 
to the midpoint 
of the opposite side. The three medians of any triangle
are concurrent (Casey 1888, p. 3), meeting in
the triangle centroid (Durell 1928) 
, which has trilinear
coordinates 
.
In addition, the medians of a triangle divide one another
in the ratio 2:1 (Casey 1888, p. 3). A median also bisects the area
of a triangle.
Let 
denote the length of the median of the 
th side 
. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
(Casey 1888, p. 23; Johnson 1929, p. 68). The area of a triangle can be expressed in terms of the medians by
 |
(3)
|
where
 |
(4)
|
See also
Bimedian, Comedian Triangles, Commandino's Theorem, Exmedian, Exmedian Point, Heronian Triangle, Medial Triangle, Median Triangle, Triangle Centroid, van Lamoen CircleExplore with Wolfram|Alpha
References
Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., 1888.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 7-8, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London, England: Macmillan, pp. 20-21, 1928.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 68, 173-175, 282-283, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London, England: Macmillian, p. 62, 1893.Referenced on Wolfram|Alpha
Triangle MedianCite this as:
Weisstein, Eric W. "Triangle Median." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleMedian.html