Median (geometry)

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that 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.

The concept of a median extends to tetrahedra.

Each median of a triangle passes through the triangle's centroid, which is the center of mass of an infinitely thin object of uniform density coinciding with the triangle.

[1] Thus, the object would balance at the intersection point of the medians.

The centroid is twice as close along any median to the side that the median intersects as it is to the vertex it emanates from.

Each median divides the area of the triangle in half, hence the name, and hence a triangular object of uniform density would balance on any median.

(Any other lines that divide triangle's area into two equal parts do not pass through the centroid.

)[2][3] The three medians divide the triangle into six smaller triangles of equal area.

, and O be the centroid (most commonly denoted G).

{\displaystyle [ADO]=[BDO],[AFO]=[CFO],[BEO]=[CEO],}

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.

{\displaystyle [AFO]=[FCO],[AFO]={\frac {1}{2}}[ACO]={\frac {1}{2}}[ABO]=[ADO]}

{\displaystyle [AFO]=[FCO]=[DBO]=[ADO]}

{\displaystyle [AFO]=[FCO]=[DBO]=[ADO]=[BEO]=[CEO]}

In 2014 Lee Sallows discovered the following theorem:[4] The lengths of the medians can be obtained from Apollonius' theorem as:

are the sides of the triangle with respective medians

These formulas imply the relationships:[5]

Let ABC be a triangle, let G be its centroid, and let D, E, and F be the midpoints of BC, CA, and AB, respectively.

For any point P in the plane of ABC then[6]

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.

The medians from sides of lengths

[8] The medians of a right triangle with hypotenuse

Any triangle's area T can be expressed in terms of its medians

A tetrahedron is a three-dimensional object having four triangular faces.

A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median of the tetrahedron.

There are four medians, and they are all concurrent at the centroid of the tetrahedron.

[10] As in the two-dimensional case, the centroid of the tetrahedron is the center of mass.

However contrary to the two-dimensional case the centroid divides the medians not in a 2:1 ratio but in a 3:1 ratio (Commandino's theorem).