results on cyclic quadrilaterals

In euclidean or hyperbolic geometry, there will be a quadrilateral with sides of length
(r,s,t,u) if and only if the sum of any three exceeds the fourth. Algebraically, this is
equivalent to the condition B(r,s,t,u) = (r+s+t-u)(s+t+u-r)(t+u+r-s)(u+r+s-t) > 0.
This follows since, if we order the lengths so that u is the largest, then only the first
factor could be negative.

The function B appears in Brahmagupta's formula for the area of a cyclic quadrilateral
in euclidean geometry. If the sides have length (r,s,t,u) then the area A is given by
A = ¼B^½(r,s,t,u). we will refer to B as Brahmagupta's Function.

In general, a given list will correspond to infinitely many quadrilaterals. If we choose
any positive number x so that there are triangles with sides (r,s,x) and (x,t,u), then
we can place such triangles so that the sides of length x correspond. In general, it will
be possible to vary x slightly, so we get the claimed infinity of solutions.

A more interesting question arises if we ask for a convex cyclic quadrilateral.

In euclidean geometry, the condition is just that B(r,s,t,u) > 0. In other words, we can
always find a quadrilateral which is cyclic if any quadrilateral exists.

It is not obvious that such x
exist when B(r,s,t,u) > 0
Here is a proof

In hyperbolic geometry, the situation is more interesting. We shall present two proofs.
One follows on from earlier work, and is really the analogue of a euclidean proof. The
second uses a new and rather strange mapping from part of the disk to the plane.

a theorem on cyclic hyperbolic quadrilaterals
There is a convex cyclic quadrilateral with sides of length r,s,t,u if and only if
B = B(s(r),s(s),s(t),s(u)) > 0, where, for real x, s(x) denotes sinh(½x).

Of course the condition is equivalent to the existence of convex cyclic euclidean
quadrilateral with sides of euclidean length s(r),s(s),s(t),s(u). the second proof
gets this form directly, but involves some(!) preliminary work.

The mapping approach leads to further hybrid theorems in a systematic way.

Before looking further at polygons, we must discuss convexity.

hyperbolic geometry