Let $\Delta_1, \Delta_2$ and $\Delta_3$ be three triangles.
Triangle $\Delta_1$ has sides equal to $x,y$ and $l_1$. Triangle $\Delta_2$ has sides equal to $y,z$ and $l_2$, and triangle $\Delta_3$ has sides $x,z$ and $l_3$. My question is:
For known values of $l_1, l_2$ and $l_3$, what is the relation between $x,y$ and $z$?
I cannot use the law of cosines because I have no information about angles. So, I do not know how to deal with this problem.
Thanks in advance for any help!
You can recover the triangle identity for $\ell_1, \ell_2, \ell_3$, since you can glue the three triangles with the corresponding sides together and obtain a tetrahedron whose fourth side is the triangle with sides $\ell_1, \ell_2, \ell_3$.