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Wikipedia gives a generalization of the law of sines to higher dimensions, as defined in F. Eriksson, The law of sines for tetrahedra and n-simplices. However, this generalization misses an important point about the standard law of sines, which relates it to the radius of the circumcircle of the triangle.

Is there a property which generalizes this relation of the 2-dimensional law of sines? In other words: is there a constant relation of this kind that all tetrahedra inscribed in a sphere of the same radius have in common?

Craig
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  • Eriksson cites https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/CarlBAllendoerfer.pdf (generalizations of theorems about triangles, by C.B. Allendoerfer) where an elementary exposition of the law of sines for tetrahedra can be found. The following article https://www.jstor.org/stable/2691141 (On the Measure of Solid Angles, by F. Eriksson) is also worth to see. – Zurab Silagadze May 12 '16 at 08:09

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Yes. See S. Yang's 2004 paper The generalized sine law and some inequalities for simplices.

Igor Rivin
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