Question

Prove that $\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|==2(a+b+c)(ab+bc+ca-a^2-b^2-c^2)$

Answer 1

$△=\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|$

$\Rightarrow R_1=R_1+R_2+R_3$

$\Rightarrow △=\left|\begin{array}{ccc}2(a+b+c)& 2(a+b+c)& 2(a+b+c)\\ a+c& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|$

$=2(a+b+c)\left|\begin{array}{ccc}1& 1& 1\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|$

$\Rightarrow C_2=C_2-C_1;C_3=C_3-C_1$

$\Rightarrow △=2(a+b+c)\left|\begin{array}{ccc}1& 0& 0\\ c+a& b-c& b-a\\ a+b& c-a& c-b\end{array}\right|$

expanding along $C_1$

$\Rightarrow △=2(a+b+c)\times 1\left|\begin{array}{cc}b-c& b-a\\ c-a& c-b\end{array}\right|$

$=2(a+b+c)[(b-c)(c-b)-(b-a)(c-a)]$

$=2(a+b+c)[bc-c^2+bc-b^2-bc+ac-a^2+ab]$

$=2(a+b+c)(ab+bc+ca-a^2-b^2-c^2)$

Hence, proved.