I am given a plane equation $Ax+By+Cz+D=0$ and coordinates $(x,y,z)$ of a point $P$ lying on a plane. I need to determine coordinates $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of two points $P_1$ and $P_2$ which are located on a line that is passing through $P$ perpendicular to the plane. Furthermore I need $P_1$ and $P_2$ to be located at the same distance $d$ from the plane.
The illustration below shows everything better than I explained. Is it possible to find out these coordinates? I've always been bad with analytic geometry so I hope someone could put this to me simply. Thanks in advance.