If $$a, b, c$$  are the sides of a triangle, then the minimum value of $$\frac{a}{{b + c - a}} + \frac{b}{{c + a - b}} + \frac{c}{{a + b - c}}$$       is equal to

Solution :
Given expression is $$\frac{1}{2}\sum {\frac{{2a}}{{b + c - a}}} $$
$$\eqalign{ & = \frac{1}{2}\sum {\left( {\frac{{2a}}{{b + c - a}} + 1} \right) - \frac{3}{2}} \cr & = \frac{1}{2}\left( {a + b + c} \right)\sum {\frac{1}{{b + c - a}} - \frac{3}{2}} \cr & {\text{Now, as }}\left( {a + b + c} \right) = \sum {\left( {b + c - a} \right)} \cr} $$
Applying A.M. $$ \geqslant $$ H.M.
Minimum value of the expression $$ = \frac{1}{2} \times 9 - \frac{3}{2} = 3.$$