# Trisecting an Angle with a Limaçon

Trisecting an Angle with a Limaçon

The graph of a limaçon in polar coordinates is ; it can be constructed geometrically. Let be a ray with polar angle . Produce this line and determine point the so that . Then .

r=b+2cos(ϕ)

OP

ϕ

A

AP=b

r=OA=2cos(ϕ)+b

A limaçon with can be used to trisect angles. Choose the angle so that one leg intersects the limaçon at .

b=1

α

A

Since the triangle is isosceles, . Since the triangle is also isosceles, . So , or .

OSP

∠PST=2ϕ=α+∠ASP

SAP

ϕ=2∠ASP

4∠ASP=α+∠ASP

∠ASP=α/3