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The triangle medians and the centroid.. 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. . Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent
For the 1-dimensional case, the geometric median coincides with the median.This is because the univariate median also minimizes the sum of distances from the points. (More precisely, if the points are p 1, ..., p n, in that order, the geometric median is the middle point (+) / if n is odd, but is not uniquely determined if n is even, when it can be any point in the line segment between the two ...
The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order).
Commandino's theorem, named after Federico Commandino (1509–1575), states that the four medians of a tetrahedron are concurrent at a point S, which divides them in a 3:1 ratio. In a tetrahedron a median is a line segment that connects a vertex with the centroid of the opposite face – that is, the centroid of the opposite triangle.
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. [ 1 ] [ 2 ] Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva , who proved a well-known theorem about cevians which also bears his name.
median triangle: reference triangle: median triangle of the median triangle: areas: | | = | | similarity: ratios: | | | | = | | | | = | | | | = The median triangle of ...
Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the centroid of the tetrahedron. [25] In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino's theorem). The centroid of a tetrahedron is the midpoint between its Monge point and ...
The median of three vertices in a tree, showing the subtree formed by the union of shortest paths between the vertices. Every tree is a median graph. To see this, observe that in a tree, the union of the three shortest paths between pairs of the three vertices a, b, and c is either itself a path, or a subtree formed by three paths meeting at a single central node with degree three.