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Trisecting an Angle Using a Conchoid

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This Demonstration shows how Nicomedes (c. 180 BC) used a conchoid to trisect an angle.

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Let be the angle to be trisected. Let and let the perpendicular to at intersect the conchoid at . Let be the intersection of and , and let be the midpoint of . Then (in a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices; see Right Triangle (Wolfram MathWorld) for a proof). Since is on the conchoid with , , and so . That is, is isoceles and ; is also isoceles and . Because , .

Putting this together, , so .

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Contributed by: Izidor Hafner (November 2012)
Open content licensed under CC BY-NC-SA

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Reference

[1] D. E. Smith, History of Mathematics, Vol. II, New York: Dover, 1958 pp. 299–300.

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