A point and a parallelogram

 Suppose $ABCD$ is a parallelogram and $P$ is a point on the same plane as $ABCD$.

Show that the locus of points $P$ such that

$$|PA|^2 + |PB|^2 + |PC|^2 + |PD|^2 = \text{constant}$$

is either a circle or nothing, depending on the constant.

What is the radius of the circle, if the parallelogram side lengths are $L$ and $W$ and the constant is $K$?

[Solution]