Algebra Question ( possible values) If 

$$k = \dfrac{a}{b+c} = \dfrac{b}{a+c} = \dfrac{c}{b+a}$$

How many possible values of $k$ are there?

Add like this:
$$(b+c)k+(a+c)k+(b+a)k=a+b+c$$
$$k=\frac{a+b+c}{b+c+a+c+b+a}=\frac12$$
if $a+b+c\ne0$, else $a=-(b+c)\Longrightarrow\dfrac a{b+c}=-1$.
So there are $2$ possible values for $k$, because first is e.g. for $a=b=c=1$ and the second e.g. for $a=b=1, c=-2$.
