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:
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.?