We are now in a position to show that, with the obvious necessary condition B(a(1),..,a(n)) > 0, there is a convex polygon with sides of lengths a(1),..,a(n). In euclidean geometry, there is a unique convex cyclic polygon. In hyperbolic geometry, we cannot expect such a theorem in view of result 4 which shows that there will be a cyclic polygon if and only if R(s(a(1)),..,s(a(n)) > 0. This is a more stringent condition. However, we will show that there is always at least one convex cyclic, horocyclic or hypercyclic polygon.
Suppose that a(1),..a(n) is a sequence of positive numbers such
We draw a vertical segment AB of length a(n). Any circle K which
We begin with the circle on AB as diameter. The CabriJava applets
In the first, some of the A(i) lie beyond B. By dragging C to the left,
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In ther second, all of the A(i) lie on the left arc AB. By dragging C to the
The experience of these applets suggests that there is exactly one solution.
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