General Networking/Lan/Wan/controling another computer
I have a computer (#1) not attached to the Internet that does some sensitive sophisticated tasks and the computer cannot have access to the internet. I have to be very careful with any software I put on this computer. Next to that initial computer is a regular computer (#2)in my office that I can control remotely. I use logmein.
The third computer is my laptop that I take with me and I do extensive traveling. That has the same account of logmein. I want to be able to control computer #1 from anywhere from computer #3 using #2 as a vehicle to accomplish this. All three computer use windows. On computer #2 and #3 I can use any necessary software.
Well that is an interesting scenario for sure! To accomplish your task, computer #1 must either be connected to the same LAN as #2, or Directly to PC2 via their own set of cables - essentially establishing a very small separate LAN using additional network Interfaces for each. the latter could be done with Ethernet cards or perhaps even more easily using USB Ethernet adapters, which are cheap and easily setup without opening up cases, etc. Newegg and Amazon each carry a lot of options for these.
More simple would be the first strategy - where computer #1 would be on the network - BUT - you would manually configure the TCP/IP config so that No Gateway or DNS exists - just an IP and Subnet Mask. This would prevent it from talking to anything that isn't on the LAN. You could also configure it's local firewall rules to prevent connections TO it from anything but PC2, if that is a possible concern. Note: PC2 COULD change it's IP over time - from DHCP, though it's not all that likely. But this could be a good enough reason to hard configure the IP on PC2 so that this can't happen.
I would think that you would then be able to use your laptop - PC3 - to remotely control pc2 and then in turn have it take control of PC1 - via either logmein or something else like remote desktop (probably comes w/ pc1 and pc2).
An added complexity is that you'll need to configure your router/firewall at that location to allow PC3 to connect through it from the Internet to PC2. Usually this would be done as NAT and/or Port Mapping rules. Most routers have interfaces which make this relatively straight-forward.
So it can all be done but it'll probably take a bit of time, configuration, testing and maybe some tweaking so that all the packets get to/from your PC3 to PC2. The inside part is probably the easiest bit since you can test all that while right on-site. Depending on what PC3 is, it might be more tricky to have it on the 'net, effectively outside the LAN, to do your testing until it all works and not being blocked by the firewall and instead having your packets forwarded and translated into the LAN from the Internet.
I hope this helps!