Show that points
$\mathrm{A}(a, b+c), \mathrm{B}(b, c+a), \mathrm{C}(c, a+b)$ are collinear
Area of ΔABC is given by the relation,
$\Delta=\frac{1}{2}\left|\begin{array}{ccc}a & b+c & 1 \\ b & c+a & 1 \\ c & a+b & 1\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}a & b+c & 1 \\ b-a & a-b & 0 \\ c-a & a-c & 0\end{array}\right|$ (Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}$ )
$=\frac{1}{2}(a-b)(c-a)\left|\begin{array}{ccc}a & b+c & 1 \\ -1 & 1 & 0 \\ 1 & -1 & 0\end{array}\right|$
$=\frac{1}{2}(a-b)(c-a)\left|\begin{array}{ccc}a & b+c & 1 \\ -1 & 1 & 0 \\ 0 & 0 & 0\end{array}\right|$ (Applying $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}+\mathrm{R}_{2}$ )
$=0 \quad$ (All elements of $R_{3}$ are 0 )
Thus, the area of the triangle formed by points A, B, and C is zero.
Hence, the points A, B, and C are collinear.