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In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been unfruitful, and at this point I'd be happy to settle for a reference to something in the literature.

Here's the conjecture:

Let $O,V_1, \dots V_{n-2} ,f$ be an $n-$pointed convex polygon in $H^2$. Given all outer angles (those between adjacent edges of the polygon), and angles $\angle V_n O V_m$ and $\angle V_{n-2} O f$, the entire polygon is uniquely determined up to isometry.

Pictorial statement of the conjecture: Given all outer angles of an $n-$pointed convex polygon, and all $\alpha_i$, the full geometry of the shape is determined up to isometry. enter image description here

Myridium
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  • Could you draw the outer angles ? I can't find one. – Kii Jun 16 '16 at 10:32
  • "Outer angles" refers to the angles subtended by the edges of the polygon. i.e. $\angle OfV_5$ is an outer angle. – Myridium Jun 16 '16 at 10:35
  • Thanks. When I construct a polygon with the same αi but taller or smaller, the outer angles are not the same. So I can not contradict your postulate right now. http://hpics.li/3dd2019 – Kii Jun 16 '16 at 10:42
  • @Kii Indeed I'm pretty sure it's correct. Could you please tell me what software you used to draw that diagram? I made mine in Geogebra but it's very fiddly. – Myridium Jun 16 '16 at 10:51
  • Do you even need the "internal" angles? – Willemien Jun 16 '16 at 20:41
  • @Willemien Yes. – Myridium Jun 16 '16 at 20:42
  • if you proof it for 4 sided polygons and for 5 sided polygons you can prove it by induction for n-sided polygons. – Willemien Jun 28 '16 at 20:42
  • @Willemien:you may be interested this software I recently released : http://math.stackexchange.com/questions/108997/does-anyone-know-a-good-hyperbolic-geometry-software-program/1981375#1981375 – Kii Oct 23 '16 at 17:18

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