# Math Help - Complex Numbers

1. ## 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 $a,b,c \in {\Cal R}$

2. I would start by expanding all three sets of brackets. See what you can do from there...

3. 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?