If a + b + c = 0, then the value of $$\frac{1}{{\left( {a + b} \right)\left( {b + c} \right)}} + $$   $$\frac{1}{{\left( {a + c} \right)\left( {b + a} \right)}} + $$   $$\frac{1}{{\left( {c + a} \right)\left( {c + b} \right)}}$$   $$ = ?$$

Solution (By Examveda Team)

a + b + c = 0
$$\frac{1}{{\left( {a + b} \right)\left( {b + c} \right)}} + $$   $$\frac{1}{{\left( {a + c} \right)\left( {b + a} \right)}} + $$   $$\frac{1}{{\left( {c + a} \right)\left( {c + b} \right)}}$$
$$\eqalign{ & \Rightarrow \frac{{\left( {a + c} \right) + \left( {b + c} \right) + \left( {a + b} \right)}}{{\left( {a + b} \right)\left( {a + c} \right)\left( {b + c} \right)}} \cr & \Rightarrow \frac{{2\left( {a + b + c} \right)}}{{\left( {a + b} \right)\left( {a + c} \right)\left( {b + c} \right)}}{\text{ }}\left( {\because a + b + c = 0} \right) \cr & \Rightarrow 0{\text{ }} \cr} $$