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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?

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

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