[@ABC].

\[centered to thin\] If $\Gamma$ is $k$-centered hyperbolic then $\Gamma$ is $4k$-thin hyperbolic.

The following proof is very similar to the proof of an existence of a global minsize of triangles implies slim triangles in [@ABC] (Proposition 2.1).

We denote $[a,b]$ as a geodesic between $a$ and $b$; if $c \in [a,b]$ then $[a, c]$ or $[c,b]$ refers to the subpath of $[a,b]$ with $c$ as one of the endpoints. Consider the triangle $xyz$ and assume it is $k$-centered. Let $p$ be the centered point and $x'$ be the point on the edge $[y,z]$ closest to $p$. Similarly define $y'$ and $z'$. Suppose there is a point $t \in [x,z']$ such that $d(t, [x, y']) > 2k$. Let $u$ be the point in $[t, z']$ nearest to $t$ such that $d(u, u') = 2k$ for some point $u' \in [x, y']$, see Figure \[center to thin figure\].

Consider the geodesic triangle $uu'x$. There exists points $a$, $b$, and $c$ on the three sides of $uu'x$ that are less than or equal to $k$ away from some point $q$, see Figure \[center to thin figure\]. Since $a \in [x, u]$, by assumption $a$ does not lie in $[t, u]$ and 