Three circles of radii a,b and c (a<b<c) touch each other externally. If they have x-axis as a common tangent, then

 
In the figure, circle $S_1,S_2$ and $S_3$ have b, c and a, respectively.
From the figure,  $\begin{array}{l}A{C}_1=b,B{C}_2=c{\text{ and CC}}_3=a\\ D{C}_1=b-a,E{C}_2=c-a,F{C}_2=c-b\end{array}$

$\begin{array}{l}\Rightarrow \sqrt{{(b+c)}^2-{(b-c)}^2}\\ \Rightarrow \sqrt{bc}=\sqrt{ab}+\sqrt{ac}\\ \Rightarrow \frac{1}{\sqrt{a}}=\frac{1}{\sqrt{c}}+\frac{1}{\sqrt{b}}\end{array}$