Trisecting an Angle Using a Conchoid

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

Let the point be at the distance from the point O on the line , that is, . Draw a straight line through perpendicular to . Let a line through intersect the line at . On the line produced in both directions, mark and so that The locus of the points and is a conchoid with pole .

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 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 .