Why would the answer for:
When you first meet someone, you should shake them ______ the hands.
It's by and not with; the by sounds cumbersome to me, though by using the hands sounds ok to me.
While here says that with is used when tools or parts of the body mentioned.
He pushed the door with his foot.
Notice those with his foot and by the hands, how could they be different?
The site also confuse me even more by stating by hand means handmade, made by a person, not by a machine.