Question

If $2s=a+b+c,$ prove that
$\left|\begin{array}{ccc}{a}^2& {(s-a)}^2& {(s-a)}^2\\ {(s-b)}^2& b^2& {(s-b)}^2\\ {(s-c)}^2& {(s-c)}^2& c^2\end{array}\right|$
$=2s^3(s-a)(s-b)(s-c).$

Answer 1

Let $s-a=A,s-b=B,s-c=C$.
Then $A+B+C=3s-(a+b+c)=3s-2s=s$
$B+C=2s-b-c=a,C+A=b,A+B=c$
$\therefore \Delta =\left|\begin{array}{ccc}{(B+C)}^2& A^2& A^2\\ B^2& {(C+A)}^2& B^2\\ C^2& C^2& {(A+B)}^2\end{array}\right|$
It is of the form of $Q.20.$
$\therefore \Delta =2ABC(A+B+C{)}^3$.
$=2(s-a)(s-b)(s-c)s^3$.