WOLFRAM|DEMONSTRATIONS PROJECT

Trisecting an Angle with a Limaçon

angle to trisect

labels

∠ASP = α / 3

The graph of a limaçon in polar coordinates is

r=b+2cos(ϕ)

; it can be constructed geometrically. Let

OP

be a ray with polar angle

ϕ

. Produce this line and determine point the

A

so that

AP=b

. Then

r=OA=2cos(ϕ)+b

.

A limaçon with

b=1

can be used to trisect angles. Choose the angle

α

so that one leg intersects the limaçon at

A

.

Since the triangle

OSP

is isosceles,

∠PST=2ϕ=α+∠ASP

. Since the triangle

SAP

is also isosceles,

ϕ=2∠ASP

. So

4∠ASP=α+∠ASP

, or

∠ASP=α/3

.