Cayley–Menger determinant
In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a formula_1-dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger. Definition. Let formula_2 be formula_3 points in formula_4-dimensional Euclidean space, with formula_5. These points are the vertices of an "n"-dimensional simplex: a triangle when formula_6; a tetrahedron when formula_7, and so on. Let formula_8 be the distances between vertices formula_9 and formula_10. The content, i.e.  the "n"-dimensional volume of this simplex, denoted by formula_11, can be expressed as a function of determinants of certain matrices, as follows: This is the Cayley–Menger determinant. For formula_13 it is a symmetric polynomial in the formula_14's and is thus invariant under permutation of these quantities. This fails for formula_15 but it is always invariant under permutation of the vertices. Proof. Let the column vectors formula_2 be formula_3 points in formula_18-dimensional Euclidean space. Starting with the volume formula we note that the determinant is unchanged when we add an extra row and column to make an formula_20  matrix, where formula_22 is the square of the length of the vector formula_23. Additionally, we note that the formula_20 matrix has a determinant of formula_26. Thus, Generalization to hyperbolic and spherical geometry. There are spherical and hyperbolic generalizations. A proof can be found here. In a spherical space of dimension formula_28 and constant curvature formula_29, any formula_30 points satisfy where formula_32, and formula_33 is the spherical distance between points formula_34. In a hyperbolic space of dimension formula_28 and constant curvature formula_36, any formula_30 points satisfy where formula_39, and formula_33 is the hyperbolic distance between points formula_34. Example. In the case  of formula_42, we have that formula_43 is the area of a triangle and thus we will denote this by formula_44. By the Cayley–Menger determinant, where the triangle has side lengths formula_45, formula_46 and formula_47, The result in the third line is due to the Fibonacci identity. The final line can be rewritten to obtain Heron's formula for the area of a triangle given three sides, which was known to Archimedes prior. In the case of formula_49, the quantity formula_50 gives the volume of a tetrahedron, which we will denote by formula_51. For distances between formula_9 and formula_23 given  by formula_33, the Cayley–Menger determinant gives Finding the circumradius of a simplex. Given a nondegenerate "n"-simplex, it has a circumscribed "n"-sphere, with radius formula_56. Then the ("n" + 1)-simplex made of the vertices of the "n"-simplex and the center of the "n"-sphere is degenerate. Thus, we have In particular, when formula_58, this gives the circumradius of a triangle in terms of its edge lengths.