This paper studies the effects of payload inertial parameters on its consistent locations for a planar three degree-of-freedom parallel manipulator in the presence of singularities. It is analytically shown that for a given motion, the consistent locations of different payloads form in general concentric circles on the moving platform plane. A motion planning method is also introduced for planar three-degree-of-freedom parallel manipulators passing through a singular configuration with different payloads. Within these scopes, eleven propositions are stated and proved. This work is believed to be highly useful for a wide range of industrial purposes, especially by providing great flexibility and ease for pick-and-place operations.