# Trisecting an Angle Using a Conchoid

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 .

B

d

OY

OB=d

m

B

OY

O

m

A

OA

P

P'

AP=AP'=k.

P

P'

O

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 , .

YOA

k=2AO

m

A

T

N

OT

m

M

NT

MT=MN=MA

T

k=2AO

NT=k=2OA

MA=OA

ΔAOM

∠AOM=∠AMO

ΔATM

∠AMO=2∠ATM

AT||OY

∠ATM=∠TOQ

Putting this together, , so .

∠AOM=2/3∠YOA

∠TOQ=1/3∠YOA