If a + b + c = 0, then the value of (a + b - c)² + (b + c - a)² + (c + a - b)² is?

Solution (By Examveda Team)

$${\left( {a + b - c} \right)^2}{\text{ + }}{\left( {b + c - a} \right)^2}$$     $${\text{ + }}{\left( {c + a - b} \right)^2}$$
$$\eqalign{ & \Rightarrow a + b + c = 0{\text{ }}\left( {{\text{ Given}}} \right) \cr & \Rightarrow a + b = - c \cr & \Rightarrow b + c = - a \cr & \Rightarrow a + c = - b \cr} $$
$$ \Rightarrow {\left( {a + b - c} \right)^2} + {\left( {b + c - a} \right)^2}$$      $$ + {\left( {c + a - b} \right)^2}$$
$$\eqalign{ & \Rightarrow {\left( { - c - c} \right)^2}{\text{ + }}{\left( { - a - a} \right)^2}{\text{ + }}{\left( { - b - b} \right)^2} \cr & \Rightarrow {\left( { - 2c} \right)^2}{\text{ + }}{\left( { - 2a} \right)^2}{\text{ + }}{\left( { - 2b} \right)^2} \cr & \Rightarrow 4{c^2} + 4{a^2} + 4{b^2} \cr & \Rightarrow 4\left( {{a^2} + {b^2} + {c^2}} \right) \cr} $$