Show that $\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}$ for three mutually tangent circles, each tangent to a common line

Let $A$ and $B$ be tangency points to the biggest circle and to the middle circle respectively.

Also, let $O_1$, $O_2$ and $O_3$ be centers of circles with radius $r_1$, $r_2$ and $r_3$ respectively

and let $O_3K$ and $O_3M$ be perpendiculars from $O_3$ to $O_1A$ and $O_2B$ respectively.

Thus, $O_1K=r_1-r_3$, $O_1O_3=r_1+r_3$ and by the Pythagoras's theorem we obtain: $$O_3K=\sqrt{(r_1+r_3)^2-(r_1+r_3)^2}=2\sqrt{r_1r_3}.$$ Similarly $$O_3M=2\sqrt{r_2r_3}$$ and $$AB=2\sqrt{r_1r_2}$$ and since $AB=O_3K+O_3M$, we obtain: $$2\sqrt{r_1r_2}=2\sqrt{r_1r_3}+2\sqrt{r_2r_3}$$ or $$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}.$$ Done!