An extension of Feuerbach's and Luchterhand's volume relation (Q1275294)

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.