Question

Prove that the points $(a,b+c),(b,c+a)$ and $(c,a+b)$ are collinear.

Answer 1

Finding area of triangle formed by the 3 points

$\Delta =\frac{1}{2}\left|\begin{array}{ccc}a& b+c& 1\\ b& c+a& 1\\ c& a+b& 1\end{array}\right|R_1\to R_1-\to R_3$

$R_2\to R_2\to R_3$

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

$\Rightarrow \Delta =o$ ($R_1$ and $R_2$ are some $\Rightarrow d$ det =0)

$\Rightarrow$ The ponts are colinear.