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  "body": "\n## Statement\n\nLet $ABC$ be a triangle in the Euclidean space $\\mathbb{R}^n$, and let $G = \\frac{1}{3}(A + B + C)$ be its [[centroid|0X7EEWES5G]].\nThen $G$ lies on each of the three [[medians|0XW8HX8VEV]] of $ABC$.\nMore precisely, let $M_A$ be the midpoint of $B$ and $C$. Then $G = A + \\frac{2}{3}(M_A - A)$, and the same holds for the vertices $B$ and $C$.\n\n## Remarks\n\nThe formula shows that $G$ divides each median in the ratio $2 : 1$, counted from the vertex.\nThe proof is one line: $A + \\frac{2}{3}\\left(\\frac{B + C}{2} - A\\right) = \\frac{1}{3}(A + B + C)$.\nThe statement is true in each affine space over a field in which $2$ and $3$ are invertible. Mathlib states it for each simplex.\n\n## Explanations\n\n### L1 human\n\nDraw the three medians of a paper triangle. They always cross at one point. This point is two thirds of the way from each vertex to the middle of the opposite side.\n",
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