Without expanding, prove that `Delta=|(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0`

Explanation

To prove that the determinant \( \Delta = \begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0 \) without expanding, we will use row operations.

### Step-by-Step Solution:

1. **Initial Determinant**:
\[ \Delta = \begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} \]
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