In the proof of uniqueness of a polygon with given side lengths
for euclidean geometry, we showed that, if A(0),...,A(n) lay in order
on a euclidean circle K and the |A(i)A(i+1)| were fixed, then the
length |A(0)A(n)| increased with the radius of K. Although the same
strategy works in hyperbolic geometry, the proofs are harder.
Earlier, we looked at an i-lines L intersecting the disk D in K. Then
We also used arguments which involved varying a euclidean circle
In our proof, we will need some results on saccheri quadrilaterals.
a monotonicity theorem
(1) K is the intersection of an i-line with the disk,
(2) l(1),..,l(m) are fixed positive numbers, and
(3) A(0),..A(m) lie in order on the minor (or finite) arc A(0)A(m)
of K such that l(i) = d(A(i),A(i+1)), i = 1,..,m-1.
Then, as the size of K increases, L = d(A(0),A(m)) increases
continuously, and tends to l(1)+..+l(m).
We are now ready to prove the existence of convex hyperbolic polygons under the