TITLE: How to show that five points in ℝ³ are cospherical?
QUESTION [8 upvotes]: 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.

REPLY [11 votes]: When $\begin{vmatrix}
x_1^2+y_1^2+z_1^2&x_1&y_1&z_1&1\\
x_2^2+y_2^2+z_2^2&x_2&y_2&z_2&1\\
x_3^2+y_3^2+z_3^2&x_3&y_3&z_3&1\\
x_4^2+y_4^2+z_4^2&x_4&y_4&z_4&1\\
x_5^2+y_5^2+z_5^2&x_5&y_5&z_5&1\\
\end{vmatrix}=0$.