How far from the corner?

This is one of my new favorite puzzles.

Imagine a point, P, inside a rectangle, with corners labeled A, B, C, and D in a clockwise direction. P is 6 feet from corner A, 2 feet from corner B, and 7 feet from corner C. How far is P from corner D?

I liked this puzzle. I hope I'm right.

Spoiler:

If D(0,0) then A(0,ay) B(cx,ay) C(cx,0) and the point P(px,py).
Let X=cx-px and Y=ay-py.

py^2+X^2=49
-(X^2+Y^2=4)
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py^2-Y^2=45


py^2-Y^2=45
+(px^2+Y^2=36)
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px^2+py^2=81

px^2+py^2=DP^2

DP=9.

Your answer is correct. For any point P inside the rectangle, AP^2+CP^2=BP^2+DP^2.

Interestingly, you can't determine the dimensions of the rectangle from the information given, even though there is enough information to answer the question.

Now for a tougher problem, show that the maximum possible area for the rectangle is 60. Also, if a quadrangle with sides 2, 6, 9, 7 is inscribed in a circle then it has half of this area. Finally, if the point P is allowed to wander outside the rectangle, then the minimum possible area is 24.

I used a bit of calculus to do this, but the fact that the maximum and minimum values are simple expressions and the there is this surprising relation to the inscribed quadrangle leads me to suspect that there may be some elegant, purely geometrical argument that gives the result.