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I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I managed to find was the original text of the generalization in German, and the only publication that cites it (according to Google Scholar) is some writeup in French.

Why is there no information about this theorem? Are there other generalizations of Stewart's theorem to more dimensions?

Basically, what I'm interested in is a case where you take the standard Stewart's theorem picture: standard Stewart's theorem picture

and find the relations when the side c is replaced by a triangle, then a simplex, etc. How about if you don't go to more dimensions than 3 and replace the line with a triangle, then a quadrilateral, pentagon, etc.?

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The obvious thing that comes to mind is that the "simplex" $P, A_1, \dotsc, A_n$ (where the $A_i$ are the analogues of the collinear points in standard Stewart's theorem) is degenerate, so its volume is $0.$ The volume is given by the Cayley-Menger determinant the vanishing of which gives a polynomial (quadratic, actually) relation between the $|P A_i|^2.$ Since the simplex given by the $A_i$ is itself degenerate, which indicates the vanishing of a principal minor of the Cayley-Menger matrix.

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  • $\begingroup$ Thanks for the answer. Can you expand on it a bit, though? I'm not familiar with the Cayley-Menger determinant, but why would a vanishing n-volume be important if the Stewart's theorem gives relations for the (n-1)-volume, e.g. the length of the line segments in the standard case? $\endgroup$
    – Tom D.
    Nov 24 '15 at 13:12
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    $\begingroup$ If you have $n+1$ points in $R^d,$ the Cayley Menger determinant expresses the $n$-dimensional volume of the simplex they span in terms of the squares of the pairwise distances. In the Stewart situation (the classical version), you have $4$ points in $\mathbb{R}^2,$ so the volume of the tetrahedron is obviously zero (since it is degenerate), which implies a relation between the distances. $\endgroup$
    – Igor Rivin
    Nov 24 '15 at 14:52
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I give a generalization of Stewart's theorem with two mention at here, do you give generalization of this to more dimensions?

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