WDS> With given sides occurs when the quad is inscribable in (i.e. inscribed in) a circle (and this indeed does not depend on the order of the sides due to Brahmagupta's area formula). It is also interesting that the "max area happens when quad inscribable" theorem is an "absolute" theorem, i.e. it works in both euclidean and nonEuclidean geometries. And the fact the area then is independent of the side-ordering also is "absolute" (which should be obvious). (I'm recounting all this from memory, since I recall proving this theorem a long time ago. I was going to put it in a paper I never finished.) ---------------------------- So, is the solid angle formed by four dihedrals maximal when it inscribes in a circular cone? (Actually, I should still have the max constraints. But I think the cone part was way messy.) --rwg 3 Sep 2008: "Everybody" knows that, given any triangle, the sum of the angle tangents equals their product. This "Brahmagupts" into Given any quadrilateral, the sum of the angle tangents equals their product times the sum of the cotangents. Can't be new. But probably newer than Brahmagupta, since, even for trapezoids, it just says 0=0. --rwg Am I the only one who hasn't seen In[650]:= FullSimplify[Integrate[Log[1/z]^n, {z, 0, 1}], n \[Element] Integers && n > 0] Out[650]= n! until today? It's just a change of variables in Euler's integral, but why isn't it preferred, or even mentioned, in e.g., A&S? Dicey for complex n?