If \( a, b \) and \( c \) are real numbers and \( \Delta=\left|\begin{array}{lll}b+c & c+a & a+b...

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If \( a, b \) and \( c \) are real numbers and \( \Delta=\left|\begin{array}{lll}b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a\end{array}\right|=0 \), then show that either \( a+b+c=0 \) or \( a=b=c \).
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