Angle Trisection
In this figure, notice that the three sides AB, BC, and CD are congruent.
How is the orange angle at B related to the angle at D?
Using only a compass and straightedge, we can easily construct three segments AB, BC, CD of the same length, and we can do this in such a way that C is on segment AD. However, it is not always possible to meet the extra requirement that a specific angle measurement appears in the diagram.
However, if we are allowed to mark our straightedge, a construction method called neusis becomes possible. In the figure below, it means sliding a fixed-length segment FG so that F and G remain collinear with E (all of this made possible by the marked straightedge) until G lands on the given circle.
That sets up a configuration just like the one above, and we have constructed angle EFC with one-third the measure of angle ABC.