# Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $$(a,b,c)$$ of positive real numbers, there exists a unique hyperbolic hexagon whose angles are all right angles such that in their natural circular order the sides are $$(a,x,b,y,c,z)$$ for some positive real numbers $$x,y,z$$.

I know two sources that claim to prove this. In Hubbard's book "Teichmuller theory, volume 1" he has a proof, but it rests on Exercise 3.5.5 (which he describes as "surprisingly tricky", and which I can only solve with a really terrible and unenlightening calculation). In Farb-Margalit's "Primer on mapping class groups", there is also a proof, but they essentially assert something equivalent to that exercise without proof.

Question: Does anyone know a source with a complete proof? Even better, a proof that minimizes terrible calculations.

• "Ratcliffe, Foundations of Hyperbolic Manifolds is decent. (See the section on hyperbolic trigonometry.) " quote from comment at math.stackexchange.com/questions/1076278/… Nov 1, 2021 at 1:33
• @WillJagy: I had forgotten about that one. Looking at it, it does give all the details, but via a gigantic wall of formulas (sadly, par for the course in that book). Hopefully someone will know a more enlightening way of doing it...
– Lisa
Nov 1, 2021 at 1:38
• Apparently Thurston gave a very simple "Law of Sines" for right angled hexagons, see mathoverflow.net/questions/278365/… I will check, my guess is Thurston gave a proof that I would, well, like. Nov 1, 2021 at 2:45
• library.msri.org/books/gt3m/PDF/2.pdf Nov 1, 2021 at 2:49
• Alright, Thurston concludes with a couple of diagrams after presenting the hyperbolic Laws of sines, and two cosines. Those are in my little article zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf Nov 1, 2021 at 2:58

• Sam, does it show Thurston's claim about right angled pentagons, if $A,B$ are lengths of two consecutive sides, and $D$ the side that has no common vertex with either of those two, then $\cosh D = \sinh A \sinh B \; ? \;$ This is 2.6.17 on the final page of library.msri.org/books/gt3m/PDF/2.pdf , where 2.6.18 is his Law of Sin(h)es for right angled hexagons. It ought to come from clever drawings and a little trig, I don't see it. Nov 1, 2021 at 17:53