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.

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