Distance between two line segments defined by sparse points

Question

Given a set S of line segments and an additional segment L, I'm looking for a procedure that determines which of the line segments in S is nearest to L. My first thought was to compute the distance between the points of L and the points of each line segment in S, however it can be tricky as illustrated in the following figure:

Let's say that L is the blue line, I want my method to point out that the green one is the nearest, but computing the distance between the points in such example may not provide the intended result.

Therefore my question is: which is the simplest (and correct) way to address this problem.

Thank you so much for your help.

edit: note that the lines are not always parallel, horizontal nor straight

Answer 1

I ended up figuring out an answer (maybe not the best one) sooner than I expected, but thanks for your attention anyway.

My idea requires that we add a point to the beginning and end of each line, which is acceptable to my problem at hand. Then we can take each pair (L,s_i), s_i = s1,s2,..sn in S and for each one connect each point to form a polygon. Then we can simply compute its perimeter or area and return the lowest one found.

e.g.