Gregorac, Robert J. An extension of Feuerbach’s and Luchterhand’s volume relation. (English) Zbl 0917.51021 Geom. Dedicata 73, No. 1, 79-84 (1998). If \(A,B,C,D\) are pairwise distinct points in counterclockwise order on a circle \(S\subset\mathbb{R}^2\) and \(\mathbb{Q}\) is a further point in \(\mathbb{R}^2\), then (due to Feuerbach and Luchterhand) the areas of the triangles in \(ABCD\) are related by \[ (QA)^2\Delta BCD+(QC)^2\Delta ABD=(QB)^2\Delta ACD+(QD)^2\Delta ABC, \] and analogous results are also known for \(m\geq n+2\) points on the sphere \(S^{n-1}\subset\mathbb{R}^n\), where \(Q\in S^{n-1}\) is required.The author is able to omit this demand, and so he obtains a natural generalization of all such results known until now. His method is the usual inversion, based on a type of stereographic projection. Reviewer: H.Martini (Chemnitz) MSC: 51M16 Inequalities and extremum problems in real or complex geometry Keywords:Feuerbach’s formula; simplex volume; inversion; stereographic projection PDFBibTeX XMLCite \textit{R. J. Gregorac}, Geom. Dedicata 73, No. 1, 79--84 (1998; Zbl 0917.51021) Full Text: DOI