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Triangle Median


TriangleMedians

A median A_1M_1 of a triangle DeltaA_1A_2A_3 is the Cevian from one of its vertices A_1 to the midpoint M_1 of the opposite side. The three medians of any triangle are concurrent (Casey 1888, p. 3), meeting in the triangle centroid (Durell 1928) G, which has trilinear coordinates 1/a:1/b:1/c. 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 m_i denote the length of the median of the ith side a_i. Then

m_1^2=1/4(2a_2^2+2a_3^2-a_1^2)
(1)
m_1^2+m_2^2+m_3^2=3/4(a_1^2+a_2^2+a_3^2)
(2)

(Casey 1888, p. 23; Johnson 1929, p. 68). The area of a triangle can be expressed in terms of the medians by

 A=4/3sqrt(s_m(s_m-m_1)(s_m-m_2)(s_m-m_3)),
(3)

where

 s_m=1/2(m_1+m_2+m_3).
(4)

See also

Bimedian, Comedian Triangles, Commandino's Theorem, Exmedian, Exmedian Point, Heronian Triangle, Medial Triangle, Median Triangle, Triangle Centroid, van Lamoen Circle

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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: 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: 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: Macmillian, p. 62, 1893.

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Triangle Median

Cite this as:

Weisstein, Eric W. "Triangle Median." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangleMedian.html

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