Windows 7: Wired router to Wireless Router???

Can I install just a simple router to share the internet connection then extend my network with my wireless router? I will place my existing wireless router in the center of the house this way only having to run one cable.

Currently the wireless router it is on the FAR side of the house. I would like to place it in the center of the house but all the ethernet cables run to the one side and I don't want to relocate them.

I attached and image of what I am trying to do.

I would use a WiFi Range Extender like this one. This can go anywhere in the house where you have an electricity outlet. Then you do not have to move any of your current setup around.

would this repeater/extra hop slow down the network performance?

It should not, but i guess it depends on the specs of the device. There are many different models on the market. Just compare the specs with the specs of your router.