If two circles are concentric, then the sum of the squares of the distances from any point of one of them to the endpoints of any diameter of the other, is a fixed quantity.
I'm having a really hard time with this one. For starters, I know that there are two separate cases (where the diameter is in the inner circle, and then when the diameter is in the outer circle). I also know that I can use the theorem "The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides", but I can't seem to figure out exactly how to create a parallelogram from the various situations.