Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between the boundaries of proper, geodesically complete CAT(-1) spaces, we define an extension $F : X \to Y$ of $f$, called the circumcenter extension of $f$, which is shown to be a $(1, \log 2)$-quasi-isometry, which is locally $1/2$-Holder continuous. If $X, Y$ are complete, simply connected Riemannian manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$, then the circumcenter extension is a $(1, (1 - 1/b) \log 2)$-quasi-isometry. Moreover if $g : \partial Y \to \partial X$ is the inverse of $f$, then the circumcenter extensions $F : X \to Y$ and $G : Y \to X$ of $f$ and $g$ are inverses of each other, and are $\sqrt{b}$-bi-Lipschitz. This is proved by constructing a family of extensions $F_p : X \to Y, 1 \leq p \leq \infty$ of $f$, called the hyperbolic $p$-barycenter extensions of $f$, and analyzing their behaviour as $p$ tends to $\infty$. As a corollary we obtain that if two closed, negatively curved manifolds have the same marked length spectrum, then they are bi-Lipschitz homeomorphic.