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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 ...
In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors: A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter.
As seen above, medians may not be unique. If each set contains more than half the population, then some of the population is exactly equal to the unique median. The median is well-defined for any ordered (one-dimensional) data and is independent of any distance metric. The median can thus be applied to school classes which are ranked but not ...
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.
In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle.
The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of ABC. Each side of the medial triangle is called a midsegment (or midline). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle.
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.