Given any vertex of a geodesic triangle in hyperbolic space, show that the entire triangle is contained in the $3\delta$ ball centred at that vertex, where $\delta$ is the constant of hyperbolicity.
Now since it's a geodesic triangle in hyperbolic space then the vertex $x$ is at most $\delta$ away from one of the sides, let that point on the side be $p$. Therefore $d(x,p)=\delta$.
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