  Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between
boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe
an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$,
which is constructed using circumcenters of expanding sets. The extension
$\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension
constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$
are complete, simply connected manifolds with sectional curvatures $K$
satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension
$\hat{f} : X \to Y$ is a $(1, (1 - \frac{1}{b})\log 2)$-quasi-isometry.
Circumcenter extension of Moebius maps is natural with respect to composition
with isometries.
