Trisection by Sliding a Line

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 .

THINGS TO TRY

SNAPSHOTS

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DETAILS

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.
References
[1] P. Berloquin, The Garden of the Sphinx, New York: Scribner's, 1985 p. 179.
[2] 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.
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