expanding hypercircles

 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 K may be a hyperbolic line or circle, a horocycle or a hypercircle. We observed that, if we choose C on K and map it to O by f, a hyperbolic transformation, then the euclidean radius of f(K) is independent of C and f. The radius is infinite if K is a hyperbolic line. We also used arguments which involved varying a euclidean circle through two fixed points. If we choose one of these as O, then the picture becomes simpler, and we may talk of the larger circle. The above remarks make it clear that we may transform either or both of the circles independently so chosen point on each becomes the centre. The euclidean radii will not change, so we still talk of the larger circle without ambiguity, and use the euclidean radii in our calculations. In our proof, we will need some results on saccheri quadrilaterals. a monotonicity theorem Suppose that (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 weakest possible condition, that the length of any side is less than the sum of the lengths of the others. Indeed, we can even choose a convex polygon whose vertices lie on an i-line, as we shall see.