How are the three plane equations derived in the NVIDIA's CG Tutorial chapter 8?
In this context, deriving the plane equations does not refer to the equations themselves, but to the coefficients $A$, $B$, $C$ and $D$.

The equations are simply plane equations for a plane defined by three points, where the first plane is defined by the three points $(x_i,s_i,t_i)_{i=1..3}$, the second plane is defined by $(y_i,s_i,t_i)_{i=1..3}$ and the third plane by $(z_i,s_i,t_i)_{i=1..3}$. They can be seen as auxiliary objects in deriving equations for $x$, $y$ and $z$ in terms of $s$ and $t$.