Three circles of radii a, b, c, (a < b < c) touch each other...

We have,

$LM = PN = \sqrt{{\left(a + b\right)}^{2} - {\left(b - a\right)}^{2}} = 2\sqrt{ab}$

$NQ = MS = \sqrt{{\left(a + c\right)}^{2} - {\left(c - a\right)}^{2}} = 2\sqrt{ac}$

$PQ = ST = \sqrt{{\left(b + c\right)}^{2} - {\left(c - b\right)}^{2}} = 2\sqrt{bc}$

$∵$ $PQ = PN + NQ$

$⇒ 2\sqrt{bc} = 2\sqrt{ab} + 2\sqrt{ac}$

$⇒ \dfrac{1}{\sqrt{a}} = \dfrac{1}{\sqrt{b}} + \dfrac{1}{\sqrt{c}}$