Angle Trisection using Limacon of Pascal |
Avni Pllana |
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" (PDF)
Avni Pllana "Approximate Angle Trisection" (PDF)
Avni Pllana "Approximate Construction of Heptagon and Nonagon" (PDF)
Avni Pllana "Three Concurrent Lines" (PDF)
Avni Pllana "A Generalization of Ceva's Theorem for Tetrahedron" (PDF)
Avni Pllana "A Generalization of the Nagel Point" (PDF)
Avni Pllana "An Interesting Triangle Center" (PDF)
Avni Pllana "A Derivation of Mollweide Equations" (PDF)
Avni Pllana "Miscellaneous Results on Tetrahedron" (PDF)
Avni Pllana "A Proof of Pythagoras' Theorem" (PDF)
Avni Pllana "Some results on tangential triangle" (PDF)
Avni Pllana "Some results on side-bisector reflected triangle" (PDF)
Avni Pllana "Circle Connections" (PDF)
Avni Pllana "All points lead to the symmedian point" (PDF)
Avni Pllana "Eagle Theorem" (PDF)
Avni Pllana "Incenter and Centroid" (PDF)
Avni Pllana "Deltoid as locus of orthopoles" (PDF)
Avni Pllana "Generalized Simson Line - Simplified Approach" (PDF)
Avni Pllana "Triangle and Lissajous" (PDF)
Avni Pllana "Three points on incircle" (PDF)
Avni Pllana "Three points on arbitrary circle" (PDF)
Avni Pllana "Some lines and planes of tetrahedron" (PDF)
Avni Pllana "An interesting triangle attractor" (PDF)
Avni Pllana "Some results on orthopolars for a given point" (PDF)
Avni Pllana "An interesting connection between Gegenbauer and Legendre polynomials" (PDF)
Avni Pllana "Determinants of nonsquare matrices" (PDF)