Etymological Musings

Do the words 'shoo' and 'shoe' share the same history?

With pets - key phrases like walk, treat, dinner, or bath stand out with meaning amid all of the random human babble with our animals.

Did we go from tossing a shoe at a howling dog (and by shoe I mean a soft, harmless slipper) to holding up a shoe and threatening to throw it, to simply saying the word 'shoo'?

The word origins seem to support my theory -
Shoo is German - Shcu
and
Shoe is also German - Schuh

The real problem with this is, who says "I'm going to throw this shoe if you don't shut up"? Or slings a shoe while simultaneously yelling "Shoe!"

Hm.