HMMT 2011 T4

Let $a, b$ , and $c$ be complex numbers such that $|a|=|b|=|c|=|a+b+c|=1$ . If $|a-b|=|a-c|$ and $b \neq c$ , prove that $|a+b||a+c|=2$ .

Dataset text NuminaMath olympiads. Quoted with source.

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