Metric Geometry/Triangles and δ-centers

Definition (triangle):

Let $(M,d)$ be a metric space. A triangle in $M$ is a triple $\alpha ,\beta ,\gamma :[0,1]\to M$ of continuous functions from the unit interval to $M$ so that $\alpha (1)=\beta (0)$, $\beta (1)=\gamma (0)$ and $>\gamma (1)=\alpha (0)$.

Definition ($\delta$-center):

Let $\delta >0$. Let $M$ be a metric space with metric $d$, and let $(\alpha ,\beta ,\gamma )$ be a triangle in $M$. A $\delta$-center for $(\alpha ,\beta ,\gamma )$ is a point $x_{0}\in M$ so that for all $t\in [0,1]$ we have $d(x_{0},\alpha (t))<\delta$ and $d(x_{0},\beta (t))<\delta$ and $d(x_{0},\gamma (t))<\delta$.