Question

If $x=\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}$, then find the value of $x$.

Answer 1

Given, $x=\frac{a}{b+c}=\frac{b}{a+c}=\frac{c}{a+b}$

$\Rightarrow x+1=\frac{a}{b+c}+1=\frac{b}{a+c}+1=\frac{c}{a+b}+1$

$\Rightarrow x+1=\frac{a+b+c}{b+c}=\frac{b+a+c}{a+c}=\frac{c+a+b}{a+b}$

$\Rightarrow \frac{1}{x+1}=\frac{a+b}{a+b+c}=\frac{a+c}{a+b+c}=\frac{a+b}{a+b+c}$

$\Rightarrow \frac{a+b+c}{x+1}=a+b=a+c=a+b$

which gives $a=b=c$

So from first equality, we have

$(a+a+a)=(x+1)(a+a)$

$\Rightarrow 3a=(x+1)\left(2a\right)$

$\Rightarrow x+1=\frac{3}{2}$

$\Rightarrow x=\frac{1}{2}$