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Math Help - Complex Numbers

  1. #1
    Junior Member mathfun's Avatar
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    Complex Numbers

    Let a,b,c be complex numbers such that:

    \displaystyle{(a+b)(a+c)=b,}

    \displaystyle{(b+c)(b+a)=c,}

    \displaystyle{(c+a)(c+b)=a.}

    Prove that $$a,b,c \in {\Cal R}$$
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  2. #2
    MHF Contributor
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    I would start by expanding all three sets of brackets. See what you can do from there...
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  3. #3
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    Let's dissect the question a little bit.

    If we expand those brackets, we find:

    a^2 + ac + ab + bc = b
    b^2 + ab +bc +ac = c
    c^2 + bc + ac + ab = a

    Note that all 3 equations have the same 3 terms : ab, ac and bc. So let's say "Let x = ab + ac + bc". Then our equations look more like:

    a^2 + x = b
    b^2 + x = c
    c^2 + x = a

    Does it become more clear now how to show this?
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