= Solution
At an interior point $w$ of the <geodesic segment> from $z_1$ to $z_2$, the two angles made with the segment toward $w'$ add to $\pi$. At least one is at least $\pi/2$. In the corresponding <hyperbolic triangle>, part (a) says that its opposite side $w'z_i$ is at least the adjacent side $w'w$. Therefore
$$
\boxed{d(w',w)\le\max\{d(w',z_1),d(w',z_2)\}.}
$$
If $w$ is an endpoint the assertion is immediate. The same inequality also holds when $w'$ lies on the complete <geodesic> through $z_1,z_2$: parameterize that <geodesic> by arclength and use the elementary inequality $|t-s|\le\max(|t-a|,|t-b|)$ for $a\le s\le b$. This extension will allow repeated use of the result in part (c).
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