I wrote this a couple years ago while I was taking geometry. I know it is not precise, but it is nice to see.
At last! I have proved the trisection of an angle using successive approximations!
This is set up for a graphical browser. If you don't have a graphical browser, you MAY be able to follow it, but it would be a good idea to download the picture.
Start with a given angle (ABC). Mark the rays into equal segments. Then, make a circle with the vertex (B) as the center and a side as the radius. Next, bisect the minor arc from the ends of the two sides. (A&C) You can do this by drawing a line from A to C and construct the perpendicular bisector. Next, split the arc on each side of the midpoint into 3 equal segments that intersect on the arc. If you are picky, you can do this by moving each segment of the line along a line perpendicular to it so that its endpoints are on the circle. Then you draw a line from each endpoint of the middle segment to the near endpoint of the nearest segment. Then extend the middle segment one third the length of each of the two lines you just drew and the end segments two thirds the length of their corresponding short segments.
If you wish to be more accurate, split the arc into 6 segments, then 9, etc. Then take the endpoint of the segment one third the distance from the midpoint (with 3 segments it would be the segment next to the midpoint, with 6 the second segment from the midpoint, etc.) and connect it to the vertex.
Congratulations! You have just trisected an angle using successive approximations. The exactness depends on how many segments you split the arc into.