Apply the theorem three times. For convenience place the two planets in directions labelled "east" and "west". We have a region of space where the vertical component of gravity points downward near the top of the region and points upward near the bottom. Accordingly, there is a surface where the vertical component is zero.I meant that the Great Pyrenees shares your confusion as to why that theorem was brought up in this context.
Within this surface, the north/south component of gravity is northward in the southern area and southward in the northern area. Accordingly there is a path where the north/south component is zero.
Within this line, the east/west component of gravity is eastward in the eastern area and westward in the western area. Accordingly there is a point where the east/west component is zero.