 Solution

The defining formulas imply the matching-distance identitiesFor example,and the other two follow cyclically. Thus $p,q,r$ are the tripod points of a geodesic triangle.

By the thin geodesic triangle condition, $p$ lies within $\delta$ of some point $p^{\prime }$ on $[x,y]$ or $[y,z]$. If $p^{\prime }\in [x,y]$, then the triangle inequality givesSince $q$ is the point of $[x,y]$ at distance $d(x,p)$ from $x$, the geodesic parametrization gives $d(p^{\prime },q)\le \delta$, and hence $d(p,q)\le 2\delta$. If instead $p^{\prime }\in [y,z]$, comparison of distances from $z$ gives $d(p,r)\le 2\delta$.

Applying the same argument cyclically, each of $p,q,r$ lies within $2\delta$ of at least one of the other two. The graph on these three points whose edges join pairs at distance at most $2\delta$ therefore has no isolated vertex, so it is connected. Any two vertices are joined by at most two edges, and the triangle inequality yields