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Assume I have an $n$-dimensional simplex on the points $x_0, ..., x_n$ where each $x_i \in \mathbb{R}^n$. I would like to obtain the parameters (center and radius) of it's circumscribed n-dimensional hypershpere. How do I do that?

I found the formulas for circle and ordinary shpere, but I am not sure how to generalize that properly.

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The circumradius is the ratio of two determinants, one of which is the Cayley-Menger determinant, see paragraph 9.7.3.7 in

Berger, Marcel, Geometry. I, II. Transl. from the French by M. Cole and S. Levy, Universitext. Berlin etc.: Springer-Verlag. I: XIII, 428 p.; DM 74.00; II: X, 406 p.; DM 74.00 (1987). ZBL0606.51001.

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