Xét ta có:
+) \(\left(a+b+c\right)^2=a^2+b^2+c^2+2ac+2ab+2bc\)
+) \(\left(a-b-c\right)^2=a^2+b^2+c^2-2ac-2ab+2bc\)
+) \(\left(b-c-a\right)^2=a^2+b^2+c^2-2ab-2bc+2ac\)
+) \(\left(c-a-b\right)^2=a^2+b^2+c^2-2ac-2bc+2ab\)
Nên: \(H=\left(a+b+c\right)^2+\left(a-b-c\right)^2+\left(b-c-a\right)^2+\left(c-a-b\right)^2\)
\(H=a^2+b^2+c^2+2ac+2bc+2bc+a^2+b^2+c^2-2ac-2ab+2bc+a^2+b^2+c^2-2ab-2bc+2ac+a^2+b^2+c^2-2ac-2bc+2ab\)
\(H=\left(a^2+a^2+a^2\right)+\left(b^2+b^2+b^2\right)+\left(c^2+c^2+c^2\right)+\left(2ac-2ac+2ac-2ac\right)+\left(2ab-2ab-2ab+2ab\right)+\left(2bc+2bc-2bc-2bc\right)\)
\(H=3a^2+3b^2+3c^2+0+0+0\)
\(H=3\left(a^2+b^2+c^2\right)\)
