Angle Trisection using Limacon of Pascal

Avni Pllana

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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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