I recently had a query on a result known (correctly) as "Fasbender's Theorem". It came as a result of its being set as a topic of special study in our state high schools here, but none of the examiners could enlighten me as to its provenance. The best I could get (from one of our most senior maths teachers here) were two vague and (as it happened) incorrect references. (One might meditate on the state of history of maths in our schools!) However, by diligent search I found out a lot and am hoping that someone "out there" can tell me more.
The theorem concerns what I shall call "floppy n-gons"; these have their sides prescribed, but not their angles. We now have:
*A floppy n-gon has maximal area when its vertices are concyclic.*
The theorem was first stated by Hermann Umpfenbach in Crelle 25 (1843), pp. 184-185, together with a proof of the case n=5. (Incidentally, he uses, as standard, many formulae now quite lost, and whose proof took me some time.)
The case n=4 must have been known from antiquity. For a simple proof see Chapter 2 of Durell & Robson's Advanced Trigonometry (London: Bell, 1930). Also in 1843, Eduard Fasbender gave a less expeditious proof (Crelle 26, pp. 181-182) of this case but then went on to use it to prove the general case as a ready corollary.
Fasbender's proof of the general result is now widely available as the final paragraph of a treatment of the problem by Ivan Niven (Maxima and Minima without Calculus; MAA, 1981, pp. 236-237).
3 queries to complete the story:
1. As Niven does not name Fasbender, where are there references that do give the attribution preserved correctly in the folk-lore?
2. Does anyone have a *simple & elementary* proof that a general floppy n-gon *can always* be made to lie with its vertices on a circle?
3. In modern terms, Fasbender's theorem tells us that a finite element approach to the classical isoperimetric problem produces an optimal result. Is this known in the literature of the finite element method?