Three circle touching each other and having a common tangent Three circle with radius $r_1$, $r_2$ and $r_3$ where ($r_1$<$r_2$<$r_3$) touch each other externally . If they have a common tangent then the value of $\sqrt{\frac{r_1}{r_2}}+\sqrt{\frac{r_1}{r_3}}$=__________
My approach is illustrated in the diagram

How will I proceed from here

Length of common tangent (CT) of two touching circlesis  $L=2\sqrt{r_1r_2}$.
CT of 1 and 3 is $L_{13}=2\sqrt{r_1r_3}$, CT of 1 and 2 is $L_{12}=
\sqrt{r_1r_2}$, CT of 2 and 3 is $L_{23}=2\sqrt{r_2r_3}.$
Next, $$L_{23}=L_{12}+L_{13}\implies 2 \sqrt{r_2r_3}=2\sqrt{r_1r_3}+2 \sqrt{r_1r_2}$$
Hence we get $$\sqrt{\frac{r_1}{r_2}}+\sqrt{\frac{r_1}{r_3}}=1$$
