AHSME 1983 Problem 13

If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is $0$, then $x^2+y^2+z^2$ equals $\textbf{(A)}\ \frac{ab+ac+bc}{abc} \textbf{(B)}\ \frac{a^2+b^2+c^2}{abc} \textbf{(C)}\ \frac{(a+b+c)^2}{abc} \textbf{(D)}\ \frac{(ab+ac+bc)^2}{abc} \textbf{(E)}\ \frac{(ab)^2+(ac)^2+(bc)^2}{abc}$