1 Galois actually proved that you
cannot trisect a general angle by intersecting lines and circles. In
the Analytic
Foundations of Geometry we see that in
Theorems
1.4 and
1.6
that the coordinates of new points that one can construct from old
points by intersecting lines are rational functions of the
coordinates of the old points. In
Theorems
3.2 and
3.3,
we see that the new coordinates of new points can be computed from
the coordinates of the old points by adding only one radical to a
rational computation.
Theorem
3.13 shows that the situation is similar if we intersect two
circles. It can be shown using trigonometry and Galois theory that
the coordinates of the point where a line through the origin that
makes a 20^{o} angle with the x-axis meets the unit circle
cannot be obtained in that manner.