If $$\frac{{{a^2} + {b^2}}}{{{c^2}}}$$  = $$\frac{{{b^2} + {c^2}}}{{{a^2}}}$$  = $$\frac{{{c^2} + {a^2}}}{{{b^2}}}$$  = $$\frac{1}{k}{\text{,}}$$ $$\left( {k \ne 0} \right)$$   then k = ?

$$\eqalign{ & \frac{{{a^2} + {b^2}}}{{{c^2}}} = \frac{{{b^2} + {c^2}}}{{{a^2}}} = \frac{{{c^2} + {a^2}}}{{{b^2}}} = \frac{1}{k} \cr & {\text{Put }}a = b = c = 1 \cr & \Rightarrow \frac{{1 + 1}}{1} + \frac{{1 + 1}}{1} + \frac{{1 + 1}}{1} = \frac{1}{k} \cr & \Rightarrow 2 = 2 = 2 = \frac{1}{k} \cr & \Rightarrow k = \frac{1}{2} \cr} $$