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