  We study the set of lines that meet a fixed line and are tangent to two
spheres and classify the configurations consisting of a single line and three
spheres for which there are infinitely many lines tangent to the three spheres
that also meet the given line. All such configurations are degenerate. The path
to this result involves the interplay of some beautiful and intricate geometry
of real surfaces in 3-space, complex algebraic geometry, explicit computation
and graphics.
