Is this an identity?

$(a + b - c)^2 + (a - b + c)^2 + (a - b - c)^2 = 2a^2 + 2b^2 + 2c^2$                                                          [Page No. 77]

Expand each:

$$\begin{aligned} (a + b - c)^2 &= a^2 + b^2 + c^2 + 2ab - 2ac - 2bc \\ (a - b + c)^2 &= a^2 + b^2 + c^2 - 2ab - 2bc + 2ac \\ (a - b - c)^2 &= a^2 + b^2 + c^2 - 2ab + 2bc - 2ac \\ \text{Sum} &= 3a^2 + 3b^2 + 3c^2 + (2ab - 2ab - 2ab) + (-2ac + 2ac - 2ac) + (-2bc - 2bc + 2bc) \\ &= 3a^2 + 3b^2 + 3c^2 - 2ab - 2ac - 2bc \end{aligned}$$

This is not equal to $2a^2 + 2b^2 + 2c^2$.

Hence, it is not an identity.

Substitute $a = 1, b = 1, c = 0$:

$$\text{LHS} = 4 + 0 + 0 = 4;$$

$$\text{RHS} = 2 + 2 + 0 = 4.$$

Substitute $a = b = c = 1$:

$$\text{LHS} = 1 + 1 + 1 = 3; \text{RHS} = 6$$

$$\text{LHS}
eq \text{RHS}$$

Hence, it is not an identity.