We will discuss here about the expansion of (a ± b ± c)\(^{2}\).

(a + b + c)\(^{2}\) = {a + (b + c)}\(^{2}\) = a\(^{2}\) + 2a(b + c) + (b + c)\(^{2}\)

= a\(^{2}\) + 2ab + 2ac + b\(^{2}\) + 2bc + c\(^{2}\)

= a\(^{2}\) + b\(^{2}\) + c\(^{2}\) + 2(ab + bc + ca)

= sum of squares of a, b, c + 2(sum of the products of a, b, c taking two at a time}.

Therefore, (a – b + c)\(^{2}\) = a\(^{2}\) + b\(^{2}\) + c\(^{2}\) + 2(ac – ab – bc)

Similarly for (a – b – c)\(^{2}\), etc.

**Corollaries:**

(i) a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = (a + b + c)\(^{2}\) – 2(ab + bc + ca)

(ii) ab + bc + ca = \(\frac{1}{2}\){(a + b + c)\(^{2}\) – (a\(^{2}\) + b\(^{2}\) + c\(^{2}\))}

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