It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses.

Also it is possible to construct straight line containing given point and parallel to given straight line by using only a ruler, with the condition that we can measure out a segment equal to any given segment.

But in this conditions there are difficulties with constructing perpendicular for given straight line. Of course it is equivalent to constructing of an isosceles triangle on given straight line by using only a ruler, with the condition that we can measure out a segment equal to any given segment.

I'm unable to do this, also I'm unable to prove that it is impossible. Does anybody know something about this topic?

domean a ruler, in the sense of neusis? $\endgroup$ – LSpice May 17 '19 at 17:12