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It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there exists a triangle with angle bisectors of length $x,y,z$ (the proof of this is very beautiful and uses Brouwer fixed point theorem).

I was wondering if there are some other results like this:

  • if $x,y,z>0$ satisfy the family of conditions $ \{ C_1,C_2,...,C_n \}$(possibly void) then there exists a triangle for which the lengths of some important lines (for eg. symmedians) are $x,y,z$.

Do you know any such results?

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Could you explain your motivation a little more? – Benoît Kloeckner Feb 28 '11 at 19:18
I tried to approach the symmedian problem, and I was wondering what other approaches are there for problems like this, because the using the approach used for bisectors I couldn't get very far, because of the relatively complicated expression of the symmedian length. I just want to know if there are other possible approaches. I tried searching the Internet, but I didn't find anything related. – Beni Bogosel Feb 28 '11 at 20:21

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