In any ` Delta A B C ,` prove that: `2(bc cosA+cacosB+a bcosC)=a^2+b^2+c^2`

Explanation

Using cosine rule `cos A=frac{b^{2}+c^{2}-a^{2}}{2 b c}, cos B=frac{c^{2}+a^{2}-b^{2}}{2 c a}, cos C=frac{a^{2}+b^{2}-c^{2}}{2 a b}`

Now, `2(b c cos A+c a cos B+a b cos C)`

`=2(b c times frac{b^{2}+c^{2}-a^{2}}{2 b c}+c a times frac{c^{2}+a^{2}-b^{2}}{2 c a}+a b times frac{a^{2}+b^{2}-c^{2}}{2 a b})`

`=2(frac{b^{2}+c^{2}-a^{2}}{2}+frac{c^{2}+a^{2}-b^{2}}{2}+frac{a^{2}+b^{2}-c^{2}}{2})`

`=2(frac{c^{2}+a^{2}+b^{2}}{2})`

`=a^{2}+b^{2}+c^{2}`