The GX280 line is notorious for having blown capacitors with odd symptoms like those you present here. There are also a couple simpler solutions that might get you going.
1) Verify that you are plugging your monitor into the proper video port, sometimes there is more than 1 and simply switching which port it is on will do it.
The problem with this is that you say that the fan is on high speed, which is an indication that the system is overheating. Generally the fan slows down after a few seconds after turning it on, right?
If that sounds like the issue you can verify it by opening up the tower using the 2 buttons on the short ends of the tower and pulling the top up from the base, it will open like a clamshell. You might need to take the leg/stand holder off which is attached with a thumbscrew on the bottom first.
Once inside, look at the capacitors, the image below will show you what they look like. The top of them, where the K is stamped, should be very flat. If they are bulged or exploded at all this is definitely your issue. If it is, the only real option is the replace the motherboard and given the age of these towers is probably more cost effective to replace the tower as a whole.
Hope this helps!