Arbitrary hyperbolic triangle

Question

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?

Thanks

Answer 1

Sometimes a good illustration can help a lot. Start by looking at the initial definitions in a Euclidean world. Drawing corners $a,b,c$ for a generic triangle, the points $p_a,p_b,p_c$ are already uniquely defined. Where are they? Here:

1. Can you describe the relevant parts of this configuration in words?
2. Would you happen to have any good idea for a suitable bound $\delta$ in the above scenario, i.e. given this specific triangle but with $q,r$ variable?
3. Can you prove it?
4. Can you adjust the proof so it still holds in hyperbolic geometry?
5. Can you perhaps even select $\delta$ such that it is independent of the triangle in question? Wikipedia might provide some suggestions (thanks to one question here, which also relates to another question here).