Given a triangle ABC and a midpoint M (of the line AB), is it possible to check whether the line CM is perpendicular to AB with a straightedge only? By this, I mean that points can be added imprecisely into any open region, or along any open line segment, and the straightedge can be used to draw the unique connecting lines (but no distances can be measured) with a programmable procedure.

I was hoping that the colinearity of some generated points in the end could check this, but now suspect that this is impossible.

This differs a little from "Steiner construction" becuase the "final constructed point" is given, so only confirmation is needed. In the problem here, I don't have the circle for the Poncelet–Steiner theorem.


1 Answer 1


Whatever points and lines you draw, a shear transformation parallel to $AB$ will preserve the collinearity of all collinear points, and will preserve the distances $AM$ and $MB$, but it will not preserve the angle of the line $CM$ with the line $AB$.

So whatever construction you perform with your straightedge, there is another (sheared) construction indistinguishable from the first one (as far as anything you can tell with a straightedge) in which the line corresponding to $CM$ in the new construction is not perpendicular to the line corresponding to $AB$.


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