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
.

Details

Isaac Newton used a limaçon to trisect an angle.

References

[1] T. Iwamoto. "Trisection Using Special Curves." (Nov 22, 2006) www.takayaiwamoto.com/Greek_Math/Trisect/Special_Curves/Special_Curves_Tri.html.

External Links

Limaçon (Wolfram MathWorld)
Limaçons as Loci and Other Polar Curves

Permanent Citation

Izidor Hafner
​
​"Trisecting an Angle with a Limaçon"​
​http://demonstrations.wolfram.com/TrisectingAnAngleWithALimacon/​
​Wolfram Demonstrations Project​
​Published: September 28, 2012