1 : \(\left(a+b+c\right)^2+a^2+b^2+c^2\)
\(=a^2+b^2+c^2+2ab+2bc+2ac+a^2+b^2+c^2\)
\(=\left(a^2+2ab+b^2\right)+\left(b^2+2bc+c^2\right)+\left(c^2+2ac+a^2\right)\)
\(=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)
2 : \(2\left(a-b\right)\left(c-b\right)+2\left(b-a\right)\left(c-a\right)+2\left(b-c\right)\left(a-c\right)\)
\(=2\left(ac-bc-ab+b^2\right)+2\left(bc-ac-ab+a^2\right)+2\left(ab-ac-bc+c^2\right)\)
\(=2ac-2bc-2ab+2b^2+2bc-2ac-2ab+2a^2+2ab-2ac-2bc+2c^2\)
\(=2a^2+2b^2+2c^2-2ac-2ab-2bc\)
\(=\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ac+a^2\right)\)
\(=\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\)
\(\)\(1.\) \(\left(a+b+c\right)^2+a^2+b^2+c^2\)
\(\Leftrightarrow a^2+b^2+c^2+2ab+2ac+2bc+a^2+b^2+c^2\)
\(\Leftrightarrow\left(a^2+2ab+b^2\right)+\left(a^2+2ac+c^2\right)+\left(b^2+2bc+c^2\right)\)
\(\Leftrightarrow\left(a+b\right)^2+\left(a+c\right)^2+\left(b+c\right)^2\)
\(2.\) \(2\left(a-b\right)\left(c-b\right)+2\left(b-a\right)\left(c-a\right)+2\left(b-c\right)\left(a-c\right)\)
\(\Leftrightarrow2ac-2ab-2bc+2b^2+2bc-2ab-2ac+2a^2+2ab-2bc-2ac+2c^2\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc\)
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