How to show that five points in ℝ³ are cospherical?

Question

There are many conditions equivalent to the cocircularity of four points on a plane, however i could not find any such lists for the three-dimensional analog. When do five points in three-dimensional space lie on one sphere?

Elementary conditions preferred.

Answer 1

Up to a circular inversion, to test five points to be cospherical is the same as testing four points to be coplanar. In terms of mutual distances, this task can be achieved by using the Cayley-Menger determinant.

Answer 2

When $\begin{vmatrix} x_1^2+y_1^2+z_1^2&x_1&y_1&z_1&1\\ x_2^2+y_2^2+z_2^2&x_2&y_2&z_2&1\\ x_3^2+y_3^2+z_3^2&x_3&y_3&z_3&1\\ x_4^2+y_4^2+z_4^2&x_4&y_4&z_4&1\\ x_5^2+y_5^2+z_5^2&x_5&y_5&z_5&1\\ \end{vmatrix}=0$.