Suppose,$\frac{\text{x}}{2\text{y}+\text{z}-\text{x}}=\frac{\text{y}}{2\text{z}+\text{x}-\text{y}}=\frac{\text{y}}{2\text{z}+\text{y}-\text{z}}=\frac{1}{\text{k}}$
$\Rightarrow\frac{\text{x}}{2\text{y }+\text{ z}-\text{x}}=\frac{1}{\text{k}},\frac{\text{y}}{2\text{z }+\text{ x}-\text{y}}=\frac{1}{\text{k}},\text{and }\frac{\text{y}}{2\text{z }+\text{ y}-\text{z}}=\frac{1}{\text{k}}$
$⟹ xk = 2y + z - x...................(1) yk = 2z + x - y .............. (2)$ and
$zk = 2x + y - z .............. (3)$ Adding $(1), (2)$ and $(3)$ we get $xk + yk + zk = 2x + 2y + 2z$
$⟹ (x + y + z) k = (x + y + z)$
$\therefore$ $k = 2$ Hence, each ratio is equal to $\frac{1}{2}$
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