## 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)].

### MSC:

 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