Comparison of hyperbolic metric and triangular ratio metric in a square

Let $K$ be a square in the plane and ${\rho }_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\mathrm{\ne }y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $\mathrm{\mid }x-y\mathrm{\mid }$ between $x$, $y\in K$ to the value ${\inf }_{z\in \mathrm{\partial }K}(\mathrm{\mid }x-z\mathrm{\mid }+\mathrm{\mid }z-y\mathrm{\mid })$. We obtain a sharp estimate for the ratio of $þ({\rho }_K(x,y)\mathrm{/}2)$ to $s_K(x,y)$.