There is a fascinating observation about that problem. If you tried to solve the problem analytically, you noticed that the formula for the cosine of the angle is of fourth order. The derivative is thus of third order. But it has a trivial solution, namely the intersection of the line with the line of the segment. Thus it is possible to express the two other solutions (on each side of the intersection) with square roots. This confirms to the fact that the solution can be constructed by compass and ruler.

Once we know the geometric solution, we see that we have to find the intersection of a parabola and a line. Here, the parabola is the locus of all points with same distance to one endpoint of the segment and the line. Again, the analytic expression has a solution expressible in square roots.