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