Angle Trisection by Straightedge and Compass Only

For many centuries since ancient times the angle trisection has remained one of the three geometric problems that could not have a clear solution by straightedge and compass. Should such a solution had been, no one would have spent time on it. Moreover, no need would have been to prove that its solution is impossible due to the third degree equation. No need would have been to prove that it is impossible to construct a great number of regular polygons which now by this solution is absolutely possible. Furthermore, this solution by putting an end to all this, should also make us wonder whether Algebra can substitute Geometry especially when in an equation exist arithmetic factors behind which, a possibly different structure of it, is well hidden.