Converse of the British Flag Theorem

Converse of the British Flag Theorem

Exist a theorem known as British Flag Theorem. It say that in a rectangle
$ABCD$ we have $PA^2 - PB^2 + PC^2 - PD^2 = 0$, for any point $P$ in the
plane.
I was thinking in a type of converse of this theorem. Given a polygon with
vertices $A_1,A_2,\ldots,A_n$, consider the function
$$f(P) = \sum_{i=1}^{n} (-1)^{i} {PA_i}^2, $$
where $P$ is a point in the plane. So I conjectured that
If a polygon with vertices $A_1,A_2,\ldots,A_n$ is such that $f(P) = 0$,
for all point $P$ in the plane, then this polygon is a rectangle.
I can to prove easily that if $n = 4$, then this is true. Somebody know
this problem? This conjecture is true? If this question is easy, but
requires a expert argument, can to give me a hint of how to solve this?