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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Delta$ be an $n$-[[simplex|YX1DJK2QAG]] in $\\R^N$ with vertices $v_0, \\ldots, v_n$.\nLet $o$ be its circumcentre, the centre of its [[circumsphere|MRC4WMVTAH]].\nLet $g = \\frac{1}{n+1}(v_0 + \\cdots + v_n)$ be its [[centroid|0Y5RHNRKJZ]].\nThe *Monge point* of $\\Delta$ is\n$$m = o + \\frac{n + 1}{n - 1} \\, (g - o).$$\n\n## Remarks\n\nFor a tetrahedron, $n = 3$ and $m = 2g - o$, the reflection of $o$ in $g$.\nIn that case $m$ is the common point of the six planes through the midpoint of an edge perpendicular to the opposite edge.\n\nFor a triangle, $n = 2$ and $m = 3g - 2o$, which is the [[orthocentre|0WB619CA0V]].\nExample: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has $o = (1, 1)$ and $g = (\\frac{2}{3}, \\frac{2}{3})$.\nThen $m = (1, 1) + 3 \\cdot (-\\frac{1}{3}, -\\frac{1}{3}) = (0, 0)$, the vertex of the right angle, which is indeed the orthocentre.\n\nNon-example: for a segment, $n = 1$, the factor $\\frac{n+1}{n-1}$ is undefined, so a $1$-simplex has no Monge point.\n\nFor a regular tetrahedron, $g = o$, so the Monge point is the centre.\n",
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