Question

Using determinants show that points $A(a,b+c),B(b,c+a)$ and $C(c,a+b)$ are collinear.

Answer 1

Given $A(a,b+c),B(b,c+a),C(c,a+b)$

Three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are collinear

if $\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& x_3& y_3\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$

$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ b+c& c+a& a+b\end{array}\right|$

$R_2\to R_2+R_3$

$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ a+b+c& a+b+c& a+b+c\\ b+c& c+a& a+b\end{array}\right|$

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

$R_1\to R_1-R_2$

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