## Line segment without a midpoint

### Task number: 3164

Find an example of a metric space, in which the midpoint of a line segment does not exist. In other words find a metric space $$(X, \rho)$$ and points $$x, y \in X$$ such that, there does not exit an $$s$$ which satisfies $$\rho(x, s) = \rho(y, s) = \frac12 \rho(x, y)$$.

• #### Resolution

Consider a metric space with two points $$X = \{1, 2\}$$, with the metric $$\rho(1, 1) = \rho(2, 2) = 0$$ and $$\rho(1, 2) = \rho(2, 1) = 1$$. Choose $$x = 1$$ a $$y=2$$. The point $$s$$ cannot exist, because in this space the length $$\frac12$$ does not exist.