In any ΔABC, prove that

Need to prove: $2(b c \cos A+c a \cos B+a b \cos C)=\left(a^{2}+b^{2}+c^{2}\right)$

Left hand side

$2(b c \cos A+c a \cos B+a b \cos C)$

$2\left(\mathrm{bc} \frac{\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}}{2 \mathrm{bc}}+\mathrm{ca} \frac{\mathrm{c}^{2}+\mathrm{a}^{2}-\mathrm{b}^{2}}{2 \mathrm{ca}}+\mathrm{ab} \frac{\mathrm{a}^{2}+\mathrm{b}^{2}-\mathrm{c}^{2}}{2 \mathrm{ab}}\right)$

$b^{2}+c^{2}-a^{2}+c^{2}+a^{2}-b^{2}+a^{2}+b^{2}-c^{2}$

$a^{2}+b^{2}+c^{2}$

Right hand side. [Proved]