be obtained from Apollonius' theorem as: where and are the sides of the triangle with respective medians and from their midpoints. These formulas imply the relationships: Other properties 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 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 with sides and medians The medians from sides of lengths and are perpendicular if and only if The medians of a right triangle with hypotenuse satisfy Any triangle's area T can be expressed in terms of its medians , and as follows. If their semi-sum is denoted by then Tetrahedron 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. 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). See also Angle bisector Altitude (triangle) Automedian triangle References External links 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 Straight lines defined for a triangle Articles containing proofs