Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged like so:
$\left|\begin{array}{ccccc} x^2+y^2+z^2 & x & y&z&1\\ x_1^2 + y_1^2 + z_1^2 & x_1 & y_1 & z_1 & 1\\ x_2^2 + y_2^2 + z_2^2 & x_2 & y_2 & z_2 & 1\\ x_3^2 + y_3^2 + z_3^2 & x_3 & y_3 & z_3 & 1\\ x_4^2 + y_4^2 + z_4^2 & x_4 & y_4 & z_4 & 1\\ \end{array}\right| = 0$
yields the equation for the sphere. Then one need only re-arrange terms into the more familiar form to find the center and radius. This works fine. My question is why.
This same approach also works for one or two dimensions. I'm guessing it also works for finding hyperspheres in higher-dimensional spaces as long as you have a corresponding number of points. But where did that determinant form come from? Is there an intuitive meaning for what that relationship is saying?