Problem
GEO-B3-M02-P023 Why a Straightedge Alone Cannot Find a Midpoint
B. New Original Problem. Only two points \(A\) and \(B\) are given. One is allowed to use only a straightedge: draw a line through two already constructed points and take the intersection of two already constructed lines. Prove that there is no universal construction of the midpoint of \(AB\) using only such operations.
C. Hint 1. Projective transformations preserve lines and intersections of lines.
D. Hint 2. Find a projective transformation that fixes \(A\) and \(B\) but moves the midpoint.
E. Full Solution.
Assume such a construction exists. Choose a projective transformation of the plane that fixes \(A\) and \(B\), but does not fix the midpoint \(M\) of \(AB\). Such a transformation exists because projective transformations of the line \(AB\) may fix two points and move a third.
Perform the supposed construction once. Then apply the chosen projective transformation to every object appearing at each step. Since a projective transformation sends lines to lines and preserves intersections, the second process is a valid execution of the same instruction starting from the same initial points \(A,B\).
But the result of the second process must be the image of the result of the first process, namely the image of \(M\), which is not \(M\). On the other hand, a universal instruction must again produce the midpoint of \(AB\), namely \(M\). This contradiction proves that a straightedge alone cannot construct the midpoint.
This is a strong problem not about construction, but about impossibility of construction. The key point is that midpoint is not a projective notion.