Alternative Constructs of the Lemniscate of Bernoulli

Until now there are at least three alternative methods of constructing lemniscate of Bernoulli. In this paper six alternative methods will be given to construct Lemniscate of Bernoulli in a simple way. Furthermore, at the end of the session will be given how to construct the incircle of Lemniscate of Bernoulli.


Introduction
Lemniscate of Bernoulli is a curve that is very often found because it resembles the number eight and the shape of the curve is used as an infinite symbol (1). Lemniscate of Bernoulli introduced by Jakob Bernoulli as a modification of an ellipse in the year 1694 in Acta Eruditorum (see [1], [2], [4] and [8]). The construction of Lemniscate of Bernoulli has been discussed in several books and journals (see [1], [9] and [10]). In 1784 James Watt [1] made a simple prop to construct the of Bernoulli using three bar linkage ( Figure 1). In 2014 A.V Akopyan [1] also discussed the construction of Lemniscate of Bernoulli whose illustration of construction was like Figure 2. In this article, we will discuss six other alternative methods to construct the Lemniscate of Bernoulli and its proof. Furthermore will be given a way of constructing the Lemniscate of Bernoulli incircle by first determining the horizontal tangent of the Lemniscate of Bernoulli.

Definition and Equation of Lemniscate of Bernoulli
Lemniscate has a definition like an ellipse. Based on several sources, Lemniscate of Bernoulli's definition can be obtained, namely: Definition 1 Lemniscate of Bernoulli is a curve such that for each points of the curve where the product of distances to foci are constant equals quarter of square of the distance between the foci (see [1], [4], and [8]).

Method 2.
In method 2, the initial step of constructing is to make a circle d with a radius of c then create a tangent circle and vertical tangent to the circle. Create two circles with a radius between the intersections of vertical tangents and tangent circles to the circle. The movement of the intersection of the two circles will form the curve of the Lemniscate of Bernoulli. Illustration of constructing the Lemniscate of Bernoulli with method 3 can be seen in the Figure 5. In the ∆EDB (in Figure 6) we have It can be proved that the curve formed fulfills the definition of 1, it is evident that the curve formed is the Lemniscate of Bernoulli.

Method 3.
Constructing the Lemniscate of Bernoulli curve in this method by making two lines with lengths p of 2c (lines F1F2 and AB) and two lines with lengths of √2 (line A 2 and B 1 ) that is connected as in   For ∆E'EF we have, Thus for ∆OHE we have, obtained by Equation (2), it can be proved that the curve formed by method 3 is the Lemniscate of Bernoulli.

Method 4.
The first step is to circle d fingers c centered at O then cut a circle through the point A which is √2 from the point O and cut the circle at the point of B and C. Create a hcircle fingers AC and center at 2 and circle k fingers AB and center at 1 . The movement of the intersection points of k and the circle h will form the curve of the Lemniscate of Bernoulli. The illustration of constructing the Lemniscate of Bernoulli with method 4 can be seen in Figure 9. based on the Definition 1 it can be proved that the curve formed by method 1 is Lemniscate Bernoulli.

Method 5.
Construction in this method is to make a circle d with radius √2 which is centered at 1 and circle through point 1 , 2 and point D on circle d. Then create a line i that is parallel to the line f so that it intersects the circle of e at the point F ( Figure 10). Will be shown the movement point G which will form the curve of the Lemniscate of Bernoulli.
For ∆ 1 (on F igure 13) hence, Then using Ptolemy's theorem, that is Figure 13) a quadrilateral that is in a circle, then the number of two pairs of adjacent sides is the same as the result of the diagonals [5].
Then based on Theorem 4 is obtained, Because 1 G × 1 G is equal to a quarter of the square of the distance between both points 1 and 2 , based on the Definition 1 it is proven that the curve is the lemniscate of Bernoulli.  Because 1 × 2 = 2 based on the Definition 1 it is proven that the curve is the lemniscate of Bernoulli.

Construction of Incircle the Lemniscate of Bernoulli
The idea of constructing the incircle the Lemniscate of Bernoulli is derived from the discussion carried out by Mashadi et al [7] concerning the development of the Cosmic Theorem by using incenter. In Lemniscate of Bernoulli there is an interesting discussion to be discussed, namely about the circle in Lemniscate of Bernoulli whose definition is based on the definition of the inner circle in the triangle.
Incircle of Lemniscate of Bernoulli can be defined as a circle that alludes to the Lemniscate of Bernoulli side on its pinpoint obtained from the horizontal tangent intersection with the minor axis of the Lemniscate of Bernoulli.
The construction of the Lemniscate of Bernoulli inner circle is obtained by determining the equation of the horizontal tangent of the Lemniscate of Bernoulli. For each horizontal tangent, a gradient equal to zero is obtained. The m gradient can be determined by the interpretation of the geometry derivative