It's all a matter of administration of the domain name. Since mycompany.com needs to be exposed externally. Your internal domain could be someting different or the same, depending on your DNS structure. I hope this isn't too confusing, but depending on your company size, some will have an Active Directory domain, which is internal. If you have a external only domain, then your internal IP address is exposed externally.
What Jeff purposed is the most flexible and easiest to manage since it's a one to one relationship. If you have 1 domain externally and 1 internally, that would be also pretty easy to implement. One ip address could be internal and one for external applications. I've done solutions and there isn't a huge difference.
Steve Schofield
Windows Server MVP - IIS
http://weblogs.asp.net/steveschofield
http://www.IISLogs.comLog archival solution
Install, Configure, Forget