I have a following exercise which I am struggling to believe is possible.
Let $T$ be an arbitrary hyperbolic triangle in the Poincaré Disc $D$, with vertices $a,b,c$ $\in$ $D$, and sides $[a,b], [a,c], [b,c]$. Let $p_a ∈ [b, c]$, $ p_b ∈ [a, c]$ and $p_c ∈ [a, b]$ be such that
$d_\mathbb D(a, p_b) = d_\mathbb D(a, p_c)$, $d_\mathbb D(b, p_a) = d_\mathbb D(b, p_c)$ and $d_\mathbb D(c, p_a) = d_\mathbb D(c, p_b)$
(a) Prove that for any points $q$ $\in$ $[a,p_c]$ and $r$ $\in$ $[a,p_b]$ with $d_D(a,q) = d_D(a,r)$, there exists a constant $\delta$ $\in$ $[0,\infty)$ such that $d_D(q,r)\leqslant \delta$.
(b) Conclude that any side of a hyperbolic triangle in the hyperbolic plane lies in the closed $\delta$-neighbourhood of the union of the other two sides.
I originally thought that choosing $\delta$ to be the maximum of $d_D(b,p_c)$ and $d_D(c,p_a)$ would work but then realised that I have assumed the point closest to $p_c$ on one of the other sides is $p_b$, when it could lie on $[b,c]$.
Do you have to find a $\delta$ that depends on just $b$, and $c$? Or any $\delta$ is fine providing it doesn't depend on $a,r$ and $q$? Or does part (a) mean that we have to show $d_D(q,r)$ is finite?