# Centroid

In geometry, the centroid or barycenter of an object [itex]X[itex] in [itex]n[itex]-dimensional space is the intersection of all hyperplanes that divide [itex]X[itex] into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of [itex]X[itex].

In physics, the centroid can, under some circumstances, coincide with an object's center of mass and also with its center of gravity. In some cases this leads to the usage of those terms interchangeably. For a centroid to coincide with the center of mass, the object should have uniform density, or the matter's distribution through the object should have certain properties, such as symmetry. For a centroid to coincide with the center of gravity, the centroid must coincide with the object's center of mass and the object must be under the influence of a uniform gravitational field.

A concave figure might have a centroid that is outside the figure itself. The centroid of a crescent, for example, lies somewhere in the central void.

The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). This point is also the triangle's center of mass if the triangle is made from a uniform sheet of material. The centroid divides each of the medians in the ratio 2:1.

The isogonal conjugate of a triangle's centroid is its symmedian point.

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