Here are some interesting construction problems that I stumbled upon.
In the following problems, constructions can only be done using an unmarked straightedge and compass.
I came across the first problem accidentally while watching my friend fold origami. However, it might have been posed long ago by some mathematician. The third problem is easier than the others.
Given a line segment of length and any positive integer . Show that, using straightedge and compass, it is always possible to divide the line into equal segments, no matter what is.
Given a line segment of length , is it possible to construct a line segment of length ?
Given triangle ABC, construct the circumcircle, and the incircle.
Given the perpendicular from A and two medians from A, B onto BC, AC respectively, reconstruct triangle ABC.