# Trisecting an Angle Using the Cycloid of Ceva

Trisecting an Angle Using the Cycloid of Ceva

The cycloid of Ceva has the polar equation . To trisect the angle , construct a line parallel to the polar axis (the positive axis). Let be the point of intersection of the cycloid and the line. Then the angle is one-third of the angle . Proof: let angle be and let the point on the axis be such that . Let be the orthogonal projection of on the line . The angle , so . Since , , . So angle equals , but .

r=1+2cos(2θ)

ABC

x

D

ABD

ABC

ABD

θ

F

x

|BE|=|EF|=1

G

F

BD

DEF=2θ

|BG|=1+cos(2θ)

|BD|=1+2cos(2θ)

|EG|=|GD|

|DF|=|EF|=1

DFH

4θ-∠EFA=3θ

∠DFH=∠CBA