Given only an angle and a straightedge, I was wondering if it is possible to construct the angle bisector. Please think of the straightedge as a line/segment rather than the normal ruler.
This problem seems easier in the projective plane (no issues regarding parallel lines). The problem could somehow be related to Pappus's Theorem, although I doubt it. There is something, however, that sounds more promising: the construction of harmonic conjugates, and we can definitively deal with that using only a straightedge.
This image taken from Wikipedia is the one that made me think that harmonic conjugates could be helpful. (Moreover, as I said before, they can be constructed with a straightedge.) Of course, I accept other methods and viewpoints ;)
Notwithstanding, please note that I do not even know if this construction is possible, so you might also find some contradictions to its constructibility.
Construction with straightlines is preserved during the affine transformation. However the bisector is not preserved, otherwise bisector in triangles would coincide with medians.