Target Display Mode via Ethernet?

I have a baseline, Mid-2010, 21.5" iMac, and have been using it primarily for Final Cut and Logic, whilst gaming on a Bootcamp partition. However, I have recently decided to build my own gaming PC. For this build, I would like to use my iMac as a display.

I know this is impossible using the Mini Dusplay Port, however, I was wondering whether it would be possible to use Target Display Mode with a signal from the Ethernet port, carried by a CAT6 cable. HDMI-Ethernet adapters exist, so I was wondering if this is actually possible?

I know it's extremely unlikely, as I could not find anything concerning this scheme after hours of scouring the Internet, but I decided I would ask here before giving up hope and buying a second display.

Thanks!

Thanks for the reply! I still don't understand why Apple couldn't have just used the same two-way port available in the 27" iMac in the smaller model, but oh well. Just wish I'd held on for a bit when I originally bought so I had the Thunderbolt port!