AHSME 1951 Problem 44

If $\frac{xy}{x+y}= a,\frac{xz}{x+z}= b,\frac{yz}{y+z}= c$, where $a, b, c$ are other than zero, then $x$ equals: $\textbf{(A)}\ \frac{abc}{ab+ac+bc} \textbf{(B)}\ \frac{2abc}{ab+bc+ac} \textbf{(C)}\ \frac{2abc}{ab+ac-bc}$ $\textbf{(D)}\ \frac{2abc}{ab+bc-ac} \textbf{(E)}\ \frac{2abc}{ac+bc-ab}$