AHSME 1983 Problem 29

A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$? $\textbf{(A)}\ 1 + \sqrt{2} \textbf{(B)}\ 2\sqrt{2} \textbf{(C)}\ 2 + \sqrt{2} \textbf{(D)}\ 3\sqrt{2} \textbf{(E)}\ 3+\sqrt{2}$