Please help me to solve the following problem:
Prove that in hyperbolic geometry there is no triangle $ABC$ such that the length of a mid-segment is half the length of the corresponding segment.
In other words: If there exist a triangle $ABC$, such that the length of a mid-segment is half the length of the corresponding segment, then it is only possible in Euclid geometry.
I think the structure of the solution must be the following: We should suppose that in hyperbolic geometry there exists triangle with mentioned property and then we should conclude that axiom about parallel lines in hyperbolic geometry fails : so there exists the line $L$ and the point outside the line $P$, such that there are 0 or 1 lines through $P$ that are parallel to $L$ and no more, thus contradicting the definition of hyperbolic geometry. I also think that I should find $L$ and $P$ by looking at $ABC$. But I am not able to move forward.
Thanks a lot for your hints and answers!
There is another possible approach: it is enough to prove that in hyperbolic geometry the length of mid segment is always less than the length of the segment.