Editing Centroid (section)

===Of a tetrahedron and {{mvar|n}}-dimensional simplex===
A [[tetrahedron]] is an object in [[three-dimensional space]] having four triangles as its [[face (geometry)|faces]]. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a ''median'', and a line segment joining the midpoints of two opposite edges is called a ''bimedian''. Hence there are four medians and three bimedians. These seven line segments all meet at the ''centroid'' of the tetrahedron.<ref>Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54</ref> The medians are divided by the centroid in the ratio <math>3:1.</math> The centroid of a tetrahedron is the midpoint between its [[Monge point]] and circumcenter (center of the circumscribed sphere). These three points define the ''Euler line'' of the tetrahedron that is analogous to the [[Euler line]] of a triangle.

These results generalize to any <math>n</math>-dimensional [[simplex]] in the following way. If the set of vertices of a simplex is <math>{v_0,\ldots,v_n},</math> then considering the vertices as [[vector (geometry)|vectors]], the centroid is

<math display=block>C = \frac{1}{n+1}\sum_{i=0}^n v_i.</math>

The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as <math>n+1</math> equal masses.