DEFINITION

Let [XYZ] denote the signed area of ΔXYZ ( positive
if the directed angle XYZ is counterclockwise ).

CONSIDER THE FOLLOWING

Let point P be in the plane of ABCD ( a convex quad-
rilateral that is not a parallelogram ). What is the
locus of of points P such that

    [APB] + [CPD] = [BPC] + [DPA] ?

Prove it.