Rich had some nice stepping methods to produce "semicyclic quadrilaterals" -- that is (if you weren't paying attention), three edge lengths A,B,C such that a quadrilateral with side lengths A,B,C,1 inscribes in a circle of diameter 1. He said:
I've ignored the unphysicalness of solutions with negative numbers.
Not at all! These are precisely the nonconvex solutions, where you read the points in the wrong order as you go around the circle. For instance,
Together with the unchanged A, the new solution A,B',C' = (1/2,-1/7,13/14).
That is to say, there exists a self-intersecting "bow-tie" quadrilateral with side-lengths 2,7,13,14 whose vertices all lie on a circle of diameter 14, and the length-2 edge is oriented oppositely from the 7 and 13. Coords of its vertices, for example, can be (-7,0), (7,0), (7/2, 7 sqrt(3)/2), (71/14, 39 sqrt(3)/2), with edge lengths 14, 7, "-2", 13, in that cyclic order. Of course, as we've mentioned, that's only one of the three noncongruent possibilities, since you get to permute the edges, eg to choose which one falls opposite the diameter. I'll have to think some to try to understand whether Rich's steppers manifest physically. Shouldn't the linear stepper, at least, be representable by some geometric construction? --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.