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Let $(a,b)$, $(c,d)$, $(e,f)$ be oppisite edges of the tetrahedron. Our teacher said there's a generalization of triangle inequality: $$ab+cd>ef$$ but I don't know how prove it, and I'm not sure whether it's true because I didn't see any similar results on the Internet. Could someone give me some advise? Thank you!

OneLamp
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    Are you familiar with Ptolemy's theorem? If so, apply that same proof (esp the triangle inequality approach), just that the points are in 3-D space instead of 2-D space. We have strict inequality because a 3-D tetrahedron cannot be concyclic (in a 2-D plane). – Calvin Lin Nov 15 '23 at 05:06
  • @CalvinLin I got it! Thanks a lot! – OneLamp Nov 15 '23 at 05:18
  • See https://math.stackexchange.com/questions/513071/tetrahedron-inequality and the questions linked there. – Gerry Myerson Nov 15 '23 at 05:43
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    @GerryMyerson, turned out this specific thing is called https://en.wikipedia.org/wiki/Ptolemy%27s_inequality as opposed to Ptolemy's Theorem, which I hadn't known either. – Will Jagy Nov 15 '23 at 18:07

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