Let $(X,d)$ be a metric space. For points $x,y \in X$ we define the metric segment between them as the following set:
$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$
We will say that a quadruplet of points $x,y,x',y' \in X$ is in a rectangular configuration if it holds that $d(x,y)=d(x',y')$, $d(x,y')=d(x',y)$ and $d(x,x')=d(y,y')$.
Assume $x,y \in X$ are points in a metric space $(X,d)$. Let $x',y' \in \left [ x,y \right ]$. My question is does $x,y,x',y'$ being in a rectangular configuration imply $\left [ x,y \right ]=\left [ x',y' \right ]$?
It seems highly improbable to me. However, I cannot seem to find a counterexample.
Note: The term rectangular configuration comes from the fact that any two pairs of corresponding sides of a rectangle are of equal length.