When I try to $ brew update I\'m getting the error:
error: Protocol https not supported or disabled in libcurl while accessing https://github.co
How can I determine the path to the curl that brew is using?
Homebrew uses /usr/bin/curl, i.e the version that ships with Mac OS X, as you can see here.
That being said, and as you precise, your problem is probably related to the version of libcurl that is linked with git and used for http:// and https://.
Perform a which git to determine which is the version you are being used (mine is installed under /usr/local).
Then scan the shared libraries used as follow:
$ otool -L /usr/local/git/libexec/git-core/git-http-push | grep curl
/usr/lib/libcurl.4.dylib
Replace /usr/local/ with the install directory that corresponds to your git.
Since the libcurl version used by your git exec lacks of HTTPS support, this will tell you what is this version and where it is installed.