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On the area of a polygon inscribed in a circle. (English) Zbl 1142.51016
If \(A\) is the area of the cyclic \(n\)-gon with side-lengths \(a_1, \dots,a_n\), and if \(t=16A^2\), then \(t\) is a zero of a polynomial \(F_n (T)\) whose coefficients are symmetric polynomials in the \(a_i\). A. F. Möbius investigated \(F_n\) and found its degree in 1828. However, the first to explicitly write down \(F_n\) for \(n=5\) and 6 was the late D. P. Robbins in [Discrete Comput. Geom. 12, No. 2, 223–236 (1994; Zbl 0806.52008)]. More work was done on the polynomials \(F_n\) by F. M. Maley, D. P. Robbins, and J. Roskies in [Adv. Appl. Math. 34, No. 4, 669–689 (2005; Zbl 1088.52005)] and by V. V. Varfolomeev in [Sb. Mat. 194, No. 3, 311–331 (2003; Zbl 1067.51013) and in Sb. Mat. 195, No. 2, 149–162 (2004; Zbl 1064.12001)]. A survey article is written by I. Pak in [Adv. Appl. Math. 34, No. 4, 690–696 (2005; Zbl 1088.52006)].
Unaware of these references, the authors of the paper under review prove that if \(n \geq 5\), then there is no formula that expresses the area of a cyclic \(n\)-gon in terms of its side-lengths using only arithmetic operations and extracting \(k\)-th roots. They do this by considering the cyclic pentagon with side-lengths 1, 1, 2, 3, 4, writing down the polynomial that defines its area, and showing that its Galois group is the unsolvable group \(S_5\). In other words, they prove that for the side-lengths 1, 1, 2, 3, 4, \(F_5\) is not solvable. However, the paper is self-contained and does not make use of the expression of \(F_5\) found by Robbins.
Appendix A of the paper deals with conditions on the positive numbers \(a_1, \dots, a_n\) that guarantee the existence of a (cyclic) \(n\)-gon whose side-lengths are these numbers. Here, the authors feel that their result is probably not new, but seem to be unaware of any references. This issue is indeed treated on p. 8 of [Z. A. Melzak’s, Invitation to Geometry. New York etc.: John Wiley & Sons, Inc. (1983; Zbl 0584.51001)], and a more rigorous treatment is given by I. Pinelis in [J. Geom. 82, No. 1–2, 156–171 (2005; Zbl 1080.52003)].

51M25 Length, area and volume in real or complex geometry
52A35 Helly-type theorems and geometric transversal theory
51M04 Elementary problems in Euclidean geometries
51M05 Euclidean geometries (general) and generalizations
12F10 Separable extensions, Galois theory