In any ` DeltaA B C` prove that `acosA+bcosB+c cosC=(8Delta^2)/(a b c)dot`

Explanation

From the cosine rule:

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

So

`a cos (A)+b cos (B)+c cos (C) =a frac{b^{2}+c^{2}-a^{2}}{2 b c}+b frac{a^{2}+c^{2}-b^{2}}{2 a c}+c frac{a^{2}+b^{2}-c^{2}}{2 a b} `

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

`=frac{a^{2} b^{2}+a^{2} c^{2}-a^{4}+a^{2} b^{2}+b^{2} c^{2}-b^{4}+a^{2} c^{2}+b^{2} c^{2}-c^{4}}{2 a b c} `

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

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

...