To divide an angle into 4 equal angles, you just bisect the bisected angles: 1/2 of 1/2 is 1/4.
Say you have an angle originating at the origin with one ray on the x axis, and the other ray anywhere, say on the line y = 3x, but it doesn't matter.
Set your compass to any width, we'll call it w, and with the point at the origin, mark equal lengths on the two lines of the angle, say at x = 5 and at (5,15), if you are using y = 3x as your other ray. Now place your compass on each of the marks you made and draw intersecting arcs.
Now connect the intersection of the arcs with the origin, and you have bisected the initial angle:
Here's how to prove it: Connect the arc intersection with the two marks, one in the x axis, one on the other ray.
Now, you have two isosceles triangles with two sides of length w, and sharing the 3rd side, the line segment connecting the origin and the arc intersection. So the two triangles are congruent, and consequently the two angles originating at the origin are equal. Consequently the angle is bisected.
Just do this again with ine of the half-angles, and you will have 1/4 of the original angle.
