A friend recently mentioned to me the following ARML problem from 1989: "A convex hexagon is inscribed in a circle. If its successive sides are 2, 2, 7, 7, 11, 11, compute the diameter of the circumscribed circle." (Anyone who wants to solve this on their own may go do so now, and come back later for the rest of my question.) Since we can reorder sides of this cyclic hexagon, the question is the same as asking for the diameter D of a circle in which you can inscribe a quadrilateral with side lengths 2,7,11,D. So maybe we should generalze from Pythagorean triangle, and say that a convex n-gon is Pythagorean if it has integral "hypotenuse" D and edges a1, a2,...,a_{n-1}, and can be inscribed in a semi-circle with diameter D. The obvious question is, what's the generalization of the Pythagorean theorem? In the quadrilateral case, the relation you get -- necessraily symmetric on the edges a_i -- turns out to be D * (a1^2 + a2^2 + a3^2) + 2 a1 a2 a3 = D^3 It's maybe more elegant-looking if you say that you're searching for rational solutions to the D=1 version, a^2 + b^2 + c^2 + 2abc = 1 but maybe not, since the LHS isn't homogeneous. Solutions include all Pythagorean triangles, by setting a3=0, and lots of solutions like 2-2-7-8, whose trigonometric meaning I'll leave you to ponder. Hmm, I suppose the regular hexagon's sol'n 1-1-1-2 also falls into this category. But the interesting solutions start with the 2-7-11-D from the ARML, then 2-9-12-16, 6-11-14-21, 1-12-22-26, 3-14-25-30, and so on. Do these come up anywhere else? And what's the relation between the edges and the hypotenuse for the n-gon case? Likely this is all well-known to those who know it, but not to me... --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.