Trisecting an Angle Using a Conchoid
This Demonstration shows how Nicomedes (c. 180 BC) used a conchoid to trisect an angle.[more]
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 .[less]
 D. E. Smith, History of Mathematics, Vol. II, New York: Dover, 1958 pp. 299–300.