Bajgonakova, G. A.; Mednykh, A. D. On Bretschneider’s formula for a spherical quadrilateral. (Russian) Zbl 1274.51022 Mat. Zamet. YAGU 19, No. 1, 3-11 (2012). In [C. A. Bretschneider, Arch. Math. 2, 225–261 (1842)], the following formula is published for the area \(S\) of a quadrangle on the Euclidean plane: \[ S^2= (p-a)(p-b)(p-c)(p-d)-abcd\cos^2\frac{A+C}{2}, \] where \(a,b,c\), and \(d\) are the lengths of the sides of the quadrangle, \(p=(a+b+c+d)/2\) is the semiperimeter, and \(A\) and \(C\) are two opposite angles of the quadrangle. (Observe that it is of no importance what pair of the opposite angles is chosen.)Using basic spherical trigonometry, the authors prove a similar formula for the area \(S\) of a spherical quadrilateral \(Q\) that reads as follows: \[ \sin^2\frac{S}{4}= \frac{\sin\tfrac{p-a}{2}\sin\tfrac{p-b}{2} \sin\tfrac{p-c}{2}\sin\tfrac{p-d}{2}}{\cos\tfrac{a}{2} \cos\tfrac{b}{2}\cos\tfrac{c}{2}\cos\tfrac{d}{2}}- \tan\frac{a}{2}\tan\frac{b}{2}\tan\frac{c}{2}\tan\frac{d}{2} \sin^2\frac{A-B+C-D}{4}, \tag{*} \] where \(a,b,c,d\) are the lengths of the sides of \(Q\), \(p=(a+b+c+d)/2\) is the semiperimeter of \(Q\), and \(A, B, C, D\) are the angles of \(Q\) listed in a cyclic order. The authors deduce several corollaries from the formula (*), e.g., they prove that, among all spherical quadrilaterals with the same lengths of the sides, the area is maximal for the one inscribed in a circle.Reviewer’s remark: In \(\S 14\) of the famous book [W. Blaschke, Kreis und Kugel. Berlin: Walter de Gruyter & Co (1956; Zbl 0070.17501)], the formula (*) is attributed to C. W. Baur [Zeitschr. f. Math. 6, 221–234 (1861)]. W. Blaschke also refers to an elegant proof of the formula (*) published by G. Hessenberg in [Schwarz-Festschrift, 76–83 (1914; JFM 45.0749.01)]. Reviewer: V. A. Alexandrov (Novosibirsk) Cited in 1 Review MSC: 51M25 Length, area and volume in real or complex geometry 51M10 Hyperbolic and elliptic geometries (general) and generalizations Keywords:stereographic projection; spherical trigonometry; Brahmagupta formula; Bretschneider theorem Citations:Zbl 0070.17501; JFM 45.0749.01 PDFBibTeX XMLCite \textit{G. A. Bajgonakova} and \textit{A. D. Mednykh}, Mat. Zamet. YAGU 19, No. 1, 3--11 (2012; Zbl 1274.51022) Full Text: Link