Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 (3) of Chapter II.9 from the book Metric spaces of non-positive curvature by Bridson and Haefliger, there exist two points $\\eta, \eta' \in \partial X$ and a point $y\in X$ such that $\angle_y (\eta,\eta') =\angle(\xi,\xi')$. Then by corollary 9.9 of the same chapter, the convex hull of two geodesic rays $c, c'$ issuing from $y$ with $c(\infty) = \eta$, $c'(\infty) = \eta'$ is isometric to a sector in $\mathbb{E}^2$ bounded by two rays meeting at the angle $\angle (\xi, \xi')$.
The above result gives geodesic rays with possibly different endpoints. So I would like to know whether we can have a stronger result that there are geodesic rays $c_1, c'_1$ issuing from some point $y_1\in X$ with endpoints $\xi,\xi'$ such that the convex hull of these rays is isometric to a flat sector. I think this should be true in general, but maybe there is a counter-example that I do not know.
