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  "body": "\n## Summary\n\nThe Cayley–Menger determinant of $n + 1$ points is a [[determinant|00KDZRW4C0]] built only from the squares of their pairwise distances.\nUp to a constant factor, it equals the square of the $n$-dimensional volume of the [[simplex|YX1DJK2QAG]] with these points as vertices.\nIt lets one compute volumes, and test whether given distances fit points of Euclidean space, without coordinates.\n[[Heron's formula|0FRBZ2X4HJ]] for the area of a triangle is its case $n = 2$.\n\n## Explanations\n\n### L1 human\n\nIf you know the six edge lengths of a tetrahedron, you know its shape, so you should be able to compute its volume.\nThe Cayley–Menger determinant does exactly this.\nPut the squared distances in a table, add a border of ones, and take the determinant.\nA suitable constant times this number is the squared volume.\nFor three points on a line the determinant is zero, since the flat triangle they form has area zero.\n",
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