From Wikipedia, the free encyclopedia - View original article

Not to be confused with Geometric median.

In geometry, a **median** of a triangle is a line segment 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. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.

Each median of a triangle passes through the triangle's centroid, which is the center of mass of an object of uniform density in the shape of the triangle.^{[1]} Thus the object would balance on any line through the centroid, including any median.

Each median divides the area of the triangle in half; hence the name. (Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.)^{[2]} The three medians divide the triangle into six smaller triangles of equal area.

Consider a triangle *ABC*. Let *D* be the midpoint of , *E* be the midpoint of , *F* be the midpoint of , and *O* be the centroid.

By definition, . Thus and , where represents the area of triangle ; these hold because in each case the two triangles have bases of equal length and share a common altitude from the (extended) base, and a triangle's area equals one-half its base times its height.

We have:

Thus, and

Since , therefore, . Using the same method, you can show that .

The lengths of the medians can be obtained from Apollonius' theorem as:

where *a*, *b* and *c* are the sides of the triangle with respective medians *m*_{a}, *m*_{b}, and *m*_{c} from their midpoints.

Thus we have the relationships:^{[3]}

The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex.

For any triangle,^{[4]}

- (perimeter) < sum of the medians < (perimeter of the triangle).

For any triangle with sides and medians ,^{[4]}

The medians from sides of lengths *a* and *b* are perpendicular if and only if ^{[5]}

The medians of a right triangle with hypotenuse *c* satisfy

We can express any triangle's area *T* in terms of its medians *m*_{a}, *m*_{b}, and *m*_{c} as follows. Denoting their semi-sum (*m _{a}* +

**^**Weisstein, Eric W. (2010).*CRC Concise Encyclopedia of Mathematics, Second Edition*. CRC Press. pp. 375–377. ISBN 9781420035223.**^**Bottomley, Henry. "Medians and Area Bisectors of a Triangle". Retrieved 27 September 2013.**^**Déplanche, Y. (1996).*Diccio fórmulas*. Medianas de un triángulo. Edunsa. p. 22. ISBN 978-84-7747-119-6. Retrieved 2011-04-24.- ^
^{a}^{b}Posamentier, Alfred S., and Salkind, Charles T.,*Challenging Problems in Geometry*, Dover, 1996: pp. 86-87. **^**Boskoff, Homentcovschi, and Suceava (2009),*Mathematical Gazette*, Note 93.15.**^**Benyi, Arpad, "A Heron-type formula for the triangle,"*Mathematical Gazette" 87, July 2003, 324–326.*

Wikimedia Commons has media related to .Median (geometry) |

- Medians and Area Bisectors of a Triangle
- The Medians at cut-the-knot
- Area of Median Triangle at cut-the-knot
- Medians of a triangle With interactive animation
- Constructing a median of a triangle with compass and straightedge animated demonstration
- Weisstein, Eric W., "Triangle Median",
*MathWorld*.