$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that \[ 3([ABC] + [A'B'C']) = [AB'C'] + [BC'A'] + [CA'B'] + [A'BC] + [B'CA] + [C'AB].\]