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  "body": "\n## Definition\n\nLet $k$ be a positive integer, and let $K$ be a field in which $k \\neq 0$. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space, and let $p_1, \\dots, p_k$ be points of $A$.\nThe centroid of $p_1, \\dots, p_k$ is the [[affine combination|0KJH6SRFZK]] with all weights equal to $\\frac{1}{k}$:\n$$G = \\frac{1}{k} p_1 + \\frac{1}{k} p_2 + \\dots + \\frac{1}{k} p_k.$$\nThe centroid of a triangle $ABC$ is the centroid $G = \\frac{1}{3}(A + B + C)$ of its vertices.\n\n## Remarks\n\nIn $\\mathbb{R}^n$, the coordinates of the centroid are the means of the coordinates of the points.\nIn $\\mathbb{R}^n$, the centroid is the point $x$ that makes the sum $\\sum_i \\|x - p_i\\|^2$ smallest.\nThe centroid of two points is their [[midpoint|0S706JBZBG]].\nThe centroid of a triangle is also the centre of mass of a uniform triangular plate.\n\n## Explanations\n\n### L1 human\n\nThe centroid is the average of the points. For the triangle with the vertices $(0, 0)$, $(6, 0)$ and $(0, 3)$, the centroid is $\\left(\\frac{0 + 6 + 0}{3}, \\frac{0 + 0 + 3}{3}\\right) = (2, 1)$. A cardboard triangle balances on a pin at this point.\n",
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