[racket] reliable path comparison?

I'm trying to compare two paths for equality; I misread normalize-path
and didn't realize that it does not do case folding, so that on
Windows,

   (equal? (build-path "C:\\")
              (build-path "c:\\"))

returns false.  What's the right way to compare paths for equality?

As a related question, what's the right way to see if one directory is
a subdirectory of another?