A Trisectrix Inspired by a Carpenter's Square
Move the point D along the red curve. The point C is chosen so that it is one unit above the x-axis and so that the line segment CD is two units long. The ray OE is a perpendicular bisector of CD (the red curve was created so that this was so). The green rays OE and OC trisect the blue angle. (Proof: DOE, COE, and COF are congruent right triangles.)