If $$\frac{a}{{b + c}}{\text{ = }}\frac{b}{{c + a}}$$   = $$\frac{c}{{a + b}}{\text{ = k,}}$$    then find the value of k is

Solution(By Examveda Team)

$$\eqalign{ & \frac{a}{{b + c}} = k \cr & \Rightarrow a = k\left( {b + c} \right)\,....(i) \cr & \frac{b}{{c + a}} = k \cr & \Rightarrow b = k\left( {c + a} \right)\,....(ii) \cr & \frac{c}{{a + b}} = k \cr & \Rightarrow c = k\left( {a + b} \right)\,....(iii) \cr & {\text{Adding (i), (ii) and (iii) we get,}} \cr & a + b + c \cr & = k\left( {2a + 2b + 2c} \right) \cr & \Rightarrow k = \frac{{a + b + c}}{{2(a + b + c)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2} \cr} $$