Concentric circles, and distances from one point to the endpoints of a diameter of the other. 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.

It's not so difficult if you make a good drawing. First draw a picture with two concentric circles in the xy plane. For easyness sake, draw one circle with radius $1$ and the other one radius $3$. In the small circle, draw a diameter that makes a 45 degree angle (with positive x-axis). Let's call the end points $A$ and $B$
Second, draw any point $C$ in the 4th quadrant on the bigger circle AND draw a point $D$ in the 2nd quadrant also on the bigger circle exactly opposite of the point in the 4th quadrant. 
Now I want you to connect these four points to form a quadrilateral.  Note that $AB$ is the diameter of the smaller circle and $CD$ the diamter of the bigger circle. The diameters bisect each other. Take a close look at the quadrilateral $ACBD$ What do you notice? How does that property of the parallelogram's sides and diagonals help you here? The other case goes similar.
