Assume circles $A, C, B$ tangent to each other and also tangent to one line at $X, Y, Z$. Let $a, b, c$ stand for the radii of circles $A, B, C$. Then $$\frac1{\sqrt{c}}=\frac1{\sqrt{a}}+\frac1{\sqrt{b}}$$

$XY=2\sqrt{ac}, YZ=2\sqrt{bc}, XZ=2\sqrt{ab}$, so $2\sqrt{ac}+2\sqrt{bc}=2\sqrt{ab}$. Dividing by $2\sqrt{abc}$ the proof follows.