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Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one determine the coordinates of the center of the circumsphere and the center of the insphere of these points, and what are the formulae for these points?

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  • $\begingroup$ The circumcircle of three of the points lies in the surface of the circumsphere, so drop perpendiculars through two of the circumcentres and intersect them. $\endgroup$
    – Peter Taylor
    Commented Aug 12 at 6:24

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