This is the law of the lever, which was proven by Archimedes using geometric reasoning.[2] It shows that if the distance a from the fulcrum to where the input force is applied (point A) is greater than the distance b from fulcrum to where the output force is applied (point B), then the lever amplifies the input force. If the distance from the fulcrum to the input force is less than from the fulcrum to the output force, then the lever reduces the input force. Recognizing the profound implications and practicalities of the law of the lever, Archimedes has been famously attributed with the quotation "Give me a place to stand and with a lever I will move the whole world."[3]
The use of velocity in the static analysis of a lever is an application of the principle of virtual work.