$
\begin{array}{l}
 \left\{ \begin{array}{l}
 (a - c)^2  + x^2  = (a + c)^2  \\
 (b - c)^2  + y^2  = (b + c)^2  \\
 (a - b)^2  + (x + y)^2  = (a + b)^2 
 \end{array} \right. \\
 \left\{ \begin{array}{l}
 a^2  - 2ac + c^2  + x^2  = a^2  + 2ac + c^2  \\
 b^2  - 2bc + c^2  + y^2  = b^2  + 2bc + c^2  \\
 a^2  - 2ab + b^2  + (x + y)^2  = a^2  + 2ab + b^2 
 \end{array} \right. \\
 \left\{ \begin{array}{l}
  - 2ac + x^2  = 2ac \\
  - 2bc + y^2  = 2bc \\
  - 2ab + (x + y)^2  = 2ab
 \end{array} \right. \\
 \left\{ \begin{array}{l}
 x^2  = 4ac \\
 y^2  = 4bc \\
 (x + y)^2  = 4ab
 \end{array} \right. \\
 \left\{ \begin{array}{l}
 x = 2\sqrt {ac}  \\
 y = 2\sqrt {bc}  \\
 x + y = 2\sqrt {ab} 
 \end{array} \right. \\
 \end{array}
$
Dus:

$
\begin{array}{l}
 2\sqrt {ab}  = 2\sqrt {ac}  + 2\sqrt {bc}  \\
 \Large\frac{{\sqrt {ab} }}{{\sqrt {abc} }} = \frac{{\sqrt {ac} }}{{\sqrt {abc} }} + \frac{{\sqrt {bc} }}{{\sqrt {abc} }} \\
 \Large\frac{1}{{\sqrt c }} = \frac{1}{{\sqrt b }} + \frac{1}{{\sqrt a }} \\
 \end{array}
$