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Category:3D graphics softwareQ:
$a(b+c)=b(c+a)$ if and only if $a=-b$ or $a=-c$
I want to prove that $a(b+c)=b(c+a)$ if and only if $a=-b$ or $a=-c$. How can I do that?
Thanks
A:
You can also prove this by noticing that it is equivalent to $a(b-c)=b(c-a)$, and expanding the square.
A:
$$(b+c)^2 = (c +a)(c +b)$$
$$b^2 +2bc + c^2 + ab +2ac+ ab = c^2 +2cd +a^2 +cb +2ab$$
$$b^2 +2bc = a^2 +cb + 2ab$$
$b^2 +2bc = (a+b)^2$
which implies $b=a-b$ or $b=a+b$ which means $b=0$ or $b=-a$ and thus $a=-b$ or $a=-c$ as well.
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