1
general::
1. Also, shake someone's hand. Clasp another's hand in greeting, farewell, or congratulation or as a sign of friendship or goodwill. For example, Stop fighting, boys; shake hands and be done with it, or You won first prize? Let me shake your hand. [Early 1500s] 2. shake hands on. Confirm a promise or bargain, as in We didn't sign a contract; we simply shook hands on our agreement. [Early 1900s]
American Heritage Idioms
2
general::
Phrase(s): shake hands [and] shake someone’s hand
to take someone’s hand and move it up and down to greet someone or mark an agreement with someone. (See also shake hands (with someone).) • David shook my hand when he greeted me. • Anne and John shook hands before their business appointment., Phrase(s): shake hands (with someone)
to clasp and shake the hand of someone as a greeting. • His hands were full, and I didn’t know whether to try to shake hands with him or not. • He put down his packages, and we shook hands.
McGrawhill's American Idioms And Phrasal Verbs