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
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