that $d(x,y)=a+b$, $d(x,z)=a+c$ and $d(y,z)=b+c$. Consider the metric tree $T_{\triangle}:=T(a,b,c)$, which consists of $3$ vertices with valency $1$ and $1$ vertices with valency 3, $3$ edges of length $a,b$ and $c$. There exists an isometry from $\{x,y,z\}$ to a subset of vertices of $T_{\triangle}$ (vertices with valency 1). Call these vertices $\{v_x,v_y,v_z\}$. This map $\{x,y,z\}\rightarrow \{v_x,v_y,v_z\}$ extends uniquely to a map $\chi_{\triangle}:\triangle \rightarrow T_{\triangle}$ such that restriction to each side of the triangle is an isometry. Denote $o_{\triangle}$ to be the central vertex of $T_{\triangle}$.

Let $\triangle$ be a geodesic triangle in a metric space $(X,d)$ and consider the map $\chi_{\triangle}:\triangle \rightarrow T_{\triangle}$ as defined above. Let $\delta\geq 0$. $\triangle$ is said to be $\delta$-thin if $p,q\in (\chi_{\triangle})^{-1}(t)$ implies $d(p,q)\leq \delta$ for all $t\in T_{\triangle}$. The diameter of $\chi_{\triangle}^{-1}(o_{\triangle})$ is denoted by insize $\triangle$.\
A geodesic metric space is called $\delta $-hyperbolic if there exists a $\delta\geq 0$ such that all geodesic triangles are $\delta$-thin. When a space is said to be hyperbolic it means that it is $\delta$-hyperbolic for some $\delta$.

These two definitions are equivalent but hyperbolicity constant may be different for two definitions. In that case one can take hyperbolicity as the maximum of hyperbolicity constants arising 