Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b} $ don't belong simultaneously to the interval $(1,2)$ I have to solve the following problem but I don't know how to :
Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b} $ don't belong simultaneously to the interval $(1,2).$
Here's what I've tried:
I worked on cases
$$\begin{cases} a \ge b \ge c \vee a \ge c \ge b \rightarrow \frac{a+b}{b+c-a}<0 \\ b \ge a \ge c \vee b \ge c \ge a \rightarrow \frac{c+a}{c+a-b}<0 \\  c \ge a \ge b \vee c \ge b \ge a \rightarrow \frac{a+b}{a+b-c}<0 \end{cases}  $$
Therefore $a,b,c$ can't be simultaneously in the interval $(1,2)$

Assume $a\leq b\leq c$
Notice $\frac{a+b}{a+b-c}\in(1,2)\implies \frac{a+b-c}{a+b}\in(\frac{1}{2},1)\implies \frac{c}{a+b}\in(0,\frac{1}{2})$, but $\frac{c}{a+b}\geq\frac{c}{2c}=\frac{1}{2}$
