
Ratio and proportion?
If we are given that:
$\displaystyle \frac{b + c  a}{1} = \frac{a + c  b}{2} = \frac{a + b  c}{3}$, how to find $\displaystyle a : b : c$?
(I was solving a trigonometry in triangles question, and I needed to find out $\displaystyle a:b:c$. I found that $\displaystyle \frac{s  a}{1} = \frac{s  b}{2} = \frac{s  c}{3}$, where s is semiperimeter. Hence, the the given equation. On second thought, is there any way to find $\displaystyle a:b$ if we know $\displaystyle \frac{k + a}{k + b} = \lambda$, where k and $\displaystyle \lambda$ are known constants?)

$\displaystyle \frac{b+ca}{1}=\frac{a+cb}{2}=\frac{a+bc}{3}=\frac{a+b+c}{6}=k$
$\displaystyle \left\{\begin{array}{ll}a+b+c=6k\\b+ca=k\end{array}\right.\Rightarrow b+c=\frac{7k}{2}\Rightarrow a=\frac{5k}{2}$
In a similar way find b and c in term of k.