Let x,y,z,t belongs to R^n where d(x,y)=||x-y||. Show that(Ptolemy's inequality): d(x,y)d(z,t)
You will find a neat proof of that here (pdf file: it's a research paper published in Pacific J. Math., so be warned that it may not be that easy to read, with some steps left for the reader to fill. The result that you want is Theorem 1, on the second page.)