(A.Myakishev, 9--11) Given triangle $ ABC$ and a ruler with two marked intervals equal to $ AC$ and $ BC$. By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ ABC$.
the fact that we have the measurements of AB and BC means the ruler is longer then AB or BC
so we take the ruler put it such that B is situated on one side of ruler
and C is siutated on the other side of the ruler .Now we draw the 2 lines we reached.
Now we repeat the operation but B is siutating on the other side of ruler and C on the other
we draw the line.THe 4 lines form 2 intersection.And we obtain a paralelogram with equal heights so a rombus
now conect the 2 verticles and get the midle bisector
do the same thing with AC and get the center of cercle.
I dont know if i corectly understood that the incenter is the centre of the big cercle so i give the solution for point I olso
Put the rulr on BC and draw a line using the other part
now put it on BA and draw another side
you obtain a rombus
conect B and the other vertex of rombus and get a bisector
do it for another angle and get the point I
P.S. Sorry if i missunderstand the condition