## Hausdorff distance again

Let $(M, d)$ be a Euclidean metric space and $X$ and $Y$ distinct non-empty subsets of $M$. Consider any elements $x$, $y$ and $z$ of $M$, where $d(x,y) = d(x,z) + d(z,y)$ (i.e., $z$ lies between $x$ and $y$). May the ratio of the Hausdorff distances between $z$ and $X$ and $z$ and $Y$ be then greater than the associated ratios for Hausdorff distances for $x$ and $y$, i.e.: $\dfrac{d(z,X)}{d(z,Y)} > max \lbrace \dfrac{d(x,X)}{d(x,Y)}, \dfrac{d(y,X)}{d(y,Y)} \rbrace$?