Write the value of the determinant

Let $\Delta=\mid \begin{array}{lll}a & 1 & b+c\end{array}$

$\begin{array}{lll}b & 1 & c+a\end{array}$

$\begin{array}{lll}c & 1 & a+b \mid\end{array}$

$=\mid a+b+c \quad 1 \quad b+c$

$\begin{array}{llll}a+b+c & 1 & c+a \\ a+b+c & 1 & a+b \mid\end{array}$      $\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}+C_{3}\right]$

$=a+b+c \mid 1 \quad 1 \quad b+c$

$\begin{array}{lll}1 & 1 & c+a \\ 1 & 1 & a+b \mid\end{array}$

$=(a+b+c) \times 0$

$=0$