Median (Triangle)

The Cevian from a Triangle's Vertex to the Midpoint of the opposite side is called a median of the Triangle. The three medians of any Triangle are Concurrent, meeting in the Triangle's Centroid (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. A median also bisects the Area of a Triangle.

Let $m_i$ denote the length of the median of the $i$th side $a_i$. Then

$${m_1}^2 = {\textstyle{1\over 4}}(2{a_2}^2+2{a_3}^2-{a_1}^2)$$ (1)

$${m_1}^2+{m_2}^2+{m_3}^2 = {\textstyle{3\over 4}} ({a_1}^2+{a_2}^2+{a_3}^2)$$ (2)

(Johnson 1929, p. 68). The Area of a Triangle can be expressed in terms of the medians by

$$A={\textstyle{4\over 3}}\sqrt{s_m(s_m-m_1)(s_m-m_2)(s_m-m_3)},$$ (3)

where

$$s_m\equiv {\textstyle{1\over 2}}(m_1+m_2+m_3).$$ (4)

A median triangle is a Triangle whose sides are equal and Parallel to the medians of a given Triangle. The median triangle of the median triangle is similar to the given Triangle in the ratio 3/4.

See also Bimedian, Exmedian, Exmedian Point, Heronian Triangle, Medial Triangle

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.

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.