Projecting image from computer that is a distance away
It's possible to amplify VGA for long runs, but since that's an analog signal it's usually not the best option. VGA over coax is probably more trouble than it's worth (you'd have to merge the many VGA channels somehow to use the single coax pin, and then reverse it).

VGA over Cat5e is fairly common. You still aren't running an analog signal over 200ft, but there's hardware to convert from VGA to a digital signal and then convert back at the other end with minimal latency. For example, this product, which also has a monitor port at the source end.

A less reliable option, but possibly less pricey if your projector supports it, is wireless projection over wi-fi. But like I said, this may be not reliable enough for theater application.