Archimedes' Method of Angle Trisection

We have already shown that it is impossible to trisect any angle by Euclidean construction. However, angle trisection can be easily done if we are allowed to use a marked ruler - a straightedge with two marks S and R on it such that RS = 1. The following applet illustrates the method discovered by Archimedes: Suppose is the angle at A to be trisected. Draw a unit circle as shown in the applet. Place the marked ruler such that S lies on the circle. Then is the required trisected angle. Question: Does this method work for any angle? If not, how would you modify Archimedes' method so that it will work for the remaining cases?