Take two spheres each of radius $a$.
Their centers are separated by distance $d_0$.
How shall we find the largest line segment among all lines joining a point on sphere $A$ to a point on sphere $B$?
My try
Just by looking at the diagram, I can guess that the largest line segment will be $d_0+2a$. However I cannot find a proof even though it may be simple.
For a point $P$ on circle centre $A$ and a point $Q$ on circle centre $B$, the triangle inequality gives us $$|PQ|\le |PA|+|AB|+|BQ|=d_o+2a.$$
This is therefore the maximum that can be attained.