A dozen of angle trisection methods using limacon of Pascal[1] are known, and one more such a method is shown in Fig.1.
Fig.1
Let 
be the given angle to be trisected. We draw line OC which bisects
, therefore
. Line OC intersects the limacon
( red loop ) at point D.
Limacon of Pascal[2] is defined in the polar coordinates by the equation
We use only the little loop of the limacon (1) , enlarged by factor 2, mirrored with respect to the y-axis, and shifted 1 unit to the right along the x-axis. Such a transformed limacon is defined in the rectangular coordinates by the following parametric equations
where 
, and 
is the bisector of the given angle 
to be
trisected.
At point D on both sides of line OC we draw lines DE and DF which make 
with line
OC . Points E and F trisect , and 
is equilateral.
In order to prove the above statements it suffices to show that 
. Let us denote
by 
, and 
by
. The Law of Sines for 
yields
From equations (2) and relation 
, follows
From (3) and (4) follows
and finally we obtain 
.
The presented trisection method can be generalized for the angle n-section, see Fig.2.
Fig.2
In Fig.2 angle 
is the half of the vertex angle of a regular n-gon. The Law of Sines for
in Fig.2 yields
From (5) and Fig.2 we obtain following equations
Next we show that equations (6) represent a special case of an epitrochoid[3]. An epitrochoid is defined by the following parametric equations
For the special case a = 1, and h = a+b, from (7) follows
Making in (8) the substitution
we obtain
Comparing (6) and (10), it follows that (6) can be obtained from (10) for 
, or
and by scaling (10) with the factor
[1] Loy, J. "Trisection of an Angle" http://www.jimloy.com/geometry/trisect.htm
[2] Eric W. Weisstein. "Limacon" http://mathworld.wolfram.com/Limacon.html
[3] Eric W. Weisstein. "Epitrochoid" http://mathworld.wolfram.com/Epitrochoid.html
See also:
Avni Pllana "Approximate Angle Trisection and N-Section" http://approxnsect.webs.com/
Avni Pllana "Approximate Angle Trisection" http://approxtrisect.webs.com/
Avni Pllana "Approximate Construction of Heptagon and Nonagon" http://heptanona.webs.com/
Avni Pllana "Three Concurrent Lines" http://threeconclines.webs.com/
Avni Pllana "A Generalization of Ceva's Theorem for Tetrahedron" http://tetraceva.webs.com/
Avni Pllana "A Generalization of the Nagel Point" http://mathnagel.webs.com/
Avni Pllana "An Interesting Triangle Center" http://interestingetc.webs.com/
Avni Pllana "A Derivation of Mollweide Equations" http://mollweide.webs.com/
Avni Pllana "Miscellaneous Results on Tetrahedron" http://misctetrahedron.webs.com/
Avni Pllana "A Proof of Pythagoras' Theorem" http://mathforum.org/kb/servlet/JiveServlet/download/129-2020419-6926265-581226/ProofPyth.pdf
Avni Pllana "Some results on tangential triangle" http://mathforum.org/kb/servlet/JiveServlet/download/129-2210946-7316515-658325/tangential_triangle.pdf
Avni Pllana "Some results on side-bisector reflected triangle" http://mathforum.org/kb/servlet/JiveServlet/download/129-2249511-7424086-673765/side_bisector_triangle1.pdf
Avni Pllana "Circle Connections" http://mathforum.org/kb/servlet/JiveServlet/download/129-2378278-7815459-759175/Circle_Connections.pdf
Avni Pllana "All points lead to the symmedian point" http://mathforum.org/kb/servlet/JiveServlet/download/129-2420508-7936656-788443/all_to_symmedian.pdf
Avni Pllana "Eagle Theorem" http://mathforum.org/kb/servlet/JiveServlet/download/129-2438335-8499541-809512/Eagle_Theorem.pdf