Trisection by Sliding a Line
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This Demonstration shows how to trisect an angle by sliding a line. Adjust the angle to trisect, , and then move point so that the point is on the line . The point is chosen so that is twice . The angle is a third of the angle .
Contributed by: Izidor Hafner (July 2012)
Open content licensed under CC BY-NC-SA
Let . Assume the line is parallel to , is perpendicular to , and . If the point is on the line , and , then triangles and are isosceles. The angles and are equal and are equal to , but their sum equals . So .
The problem goes back to ancient Greece, with contributions by Hippocrates, Archimedes, and Pappus.
 P. Berloquin, The Garden of the Sphinx, New York: Scribner's, 1985 p. 179.
 J. J. O'Connor and E. F. Robertson. "Trisecting an Angle." (Jun 21, 2012) www-history.mcs.st-and.ac.uk/HistTopics/Trisecting_an_angle.html.