Verify Archimedes’ construction for trisecting an angle.
We are allowed to construct points using a straightedge and compass. Moreover, our straightedge has an (arbitrary) unit distance marked on it. We are permitted to ‘slide’ the marked straightedge on the plane.
Suppose we have an angle of measure , where . Using our unit ruler, we can say (without loss of generality) that has length 1. Construct the circle with center and containing ; this circle intersects the opposite ray of our angle at a point with we call (again without loss of generality) . So . Now extend the segment to a line.
Now slide the straightedge so that one mark is on the line and the other is on the circle centered at . Finally, slide the straightedge (keeping the marks on the line and circle) so that the edge intersects point . Construct this segment, calling the points where the edge intersects the line and circle and , respectively, as in the following diagram.
In addition, label the angles , , and as shown.
Note that is isocoles, so that . Moreover, is isocoles. Now has measure , whence we have . Now has measure , and moreover , so that . Thus .