I explore the Mathstackchange for a simple formula that calculates the center of the tetrahedron circumsphere.
I find this Circumsphere of a tetrahedron. Two main approaches and some alternatives ways are mentioned in the above link.
I tried a way which also mentioned in MathWorld's "Circumsphere" entry. This approach is based on a calculate four $4\times 4$ determinants, $D_x$, $D_y$, $D_z$, and $a$; the last is $$ a = \left|\begin{array}{cccc} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{array}\right| $$ Then, we have a circumsphere center via using these formulae:
$$\text{centerX} = \frac{D_x}{2 \times a}, \quad \text{centerY}= \frac{D_y}{2 \times a}, \quad \text{centerZ}= \frac{D_z}{2\times a} $$
I want to know:
If the result of determinant $a$ is zero, how the center of circumsphere center is calculated?