The problem was here: https://ruffnsluff.blogspot.com/2021/08/a-point-and-parallelogram.html
We use the following facts
1) The diagonals of a parallelogram bisect each other.
2) Apollonius theorem: If A,B,C is a triangle and D is midpoint of BC, then |AB|2+|AC|2=2(|AD|2+|BD|2)
Let O be the intersection of the diagonals AC and BD of the parallelogram ABCD.
The by applying the Apollonius theorem of triangle PAC we get that
|PA|2+|PC|2=2(|AO|2+|PO|2)
Similarly
|PB|2+|PD|2=2(|BO|2+|PO|2)
Adding we get
|PA|2+|PB|2+|PC|2+|PD|2=4|PO|2+2(|AO|2+|BO|2)
Thus if the LHS is constant then so is |PO| and hence the locus of P is a circle with center O (or empty if the constant is smaller than 2(|AO|2+|BO|2)).
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