Let $p,q \in \mathbf{R}^2$ and let $C$ be a circle with diameter $pq$. Consider another circle $C'$ whose center is directly below that of $C$'s by $\delta$, and which passes through $p$ and $q$. Thus the radius of $C'$ is $R = \sqrt{\delta^2 + r^2}$. See Figure below.
At each point on $x \in C'$ the lies inside of $C$, I consider the ball centered at $x$ with radius $\min\{d(x,p), d(x,q)\}$. In the figure I've drawn three example balls in red. Let $\mathcal{B} = \cup_{x \in C', \text{ and in } C}B(x,\min\{d(x,p), d(x,q)\})$, be the union of this collection of balls. The ball centered at the point on $C'$ directly above $c$ is the largest in the collection, let it's radius be $r'$. It's distance to $c$ is $R-\delta$. Consider a ball $B(c, r'+ R - \delta)$, the ball centered at $c$ with radius large enough to contain the largest ball in the collection. Is it true that $\mathcal{B} \subset B(c, r'+ R - \delta)$, regardless of the value of $\delta$? If not, how much larger does the ball centered at $c$ need to be to contain $\mathcal{B}$?
The statement is true in the limiting cases, when $\delta = 0$ or $\delta = \infty$, but I can't seem to prove it in general.