Generating Lemoine Circles

Professor Ion Patrascu, Fratii Buzesti National College, Craiova, Romania Professor Florentin Smarandache, New Mexico University, USA

In this paper, we generalize the theorem relative to the first circle of Lemoine and thereby highlight a method to build Lemoine circles. Firstly, we review some notions and results.

Definition 1. It is called a simedian of a triangle the symmetric of a median of the triangle with respect to the internal bisector of the triangle that has in common with the median the peak of the triangle.

Proposition 1. In the triangle 퐴퐵퐶 , the cevian 퐴푆 , 푆 ∈ (퐵퐶), is a

푆퐵 퐴퐵 2 simedian if and only if = ( ) . 푆퐶 퐴퐶

For Proof, see [2].

Definition 2. It is called a simedian center of a triangle (or Lemoine point) the intersection of triangle’s simedians.

Theorem 1. The parallels to the sides of a triangle taken through the simedian center intersect the triangle’s sides in six concyclic points (the first Lemoine circle - 1873).

A Proof of this theorem can be found in [2].

Definition 3. We assert that in a scalene triangle 퐴퐵퐶 the line 푀푁, where 푀 ∈ 퐴퐵 and 푁 ∈ 퐴퐶, is an anti-parallel to 퐵퐶 if ∢푀푁퐴 ≡ ∢퐴퐵퐶.

1 Lemma 1. In the triangle 퐴퐵퐶, let 퐴푆 be a simedian, 푆 ∈ (퐵퐶). If 푃 is the middle of the segment (푀푁), having 푀 ∈ (퐴퐵) and 푁 ∈ (퐴퐶), belonging to the simedian 퐴푆, then 푀푁 and 퐵퐶 are anti-parallels.

Proof. We draw through 푀 and 푁, 푀푇 ∥ 퐴퐶 and 푁푅 ∥ 퐴퐵, 푅, 푇 ∈ (퐵퐶), see Figure 1. Let {푄} = 푀푇 ∩ 푁푅; since 푀푃 = 푃푁 and 퐴푀푄푁 is a parallelogram, it follows that 푄 ∈ 퐴푆.

Figure 1.

Thales' Theorem provides the relations:

퐴푁 퐵푅 퐴퐵 퐵퐶 = (1); = (2). 퐴퐶 퐵퐶 퐴푀 퐶푇 From (1) and (2), by multiplication, we obtain:

퐴푁 퐴퐵 퐵푅 ∙ = (3). 퐴푀 퐴퐶 푇퐶 Using again Thales' Theorem, we obtain:

퐵푅 퐴푄 푇퐶 퐴푄 = (4), = (5). 퐵푆 퐴푆 푆퐶 퐴푆

2 From these relations, we get

퐵푅 푇퐶 퐵푆 퐵푅 = (6) or = (7). 퐵푆 푆퐶 푆퐶 푇퐶

퐴푁 퐴퐵 In view of Proposition 1, the relations (7) and (3) drive to = , which 퐴퐵 퐴퐶 shows that ∆퐴푀푁~∆퐴퐶퐵, so ∢퐴푀푁 ≡ ∢퐴퐵퐶, therefore 푀푁 and 퐵퐶 are anti- parallels in relation to 퐴퐵 and 퐴퐶.

Remark.

1. The reciprocal of Lemma 1 is also valid, meaning that if 푃 is the middle of the anti-parallel 푀푁 to 퐵퐶, then 푃 belongs to the simedian from 퐴.

Theorem 2. (Generalization of Theorem 1) Let 퐴퐵퐶 be a scalene triangle and 퐾 its simedian center. We take 푀 ∈ 퐴퐾 and draw 푀푁 ∥ 퐴퐵, 푀푃 ∥ 퐴퐶, where 푁 ∈ 퐵퐾, 푃 ∈ 퐶퐾. Then:

i. 푁푃 ∥ 퐵퐶; ii. 푀푁, 푁푃 and 푀푃 intersect the sides of triangle 퐴퐵퐶 in six concyclic points.

Proof. In triangle 퐴퐵퐶, let 퐴퐴1, 퐵퐵1, 퐶퐶1 the simedians concurrent in 퐾 퐴푀 퐵푁 퐴푀 퐶푃 (see Figure 2). We have from Thales' Theorem that: = (1); = (2). 푀퐾 푁퐾 푀퐾 푃퐾 퐵푁 퐶푃 From relations (1) and (2), it follows that = (3), which shows that 푁푃 ∥ 푁퐾 푃퐾 퐵퐶. Let 푅,푆,푉,푊,푈,푇 be the intersection points of the parallels 푀푁, 푀푃, 푁푃 of the sides of the triangles to the other sides. Obviously, by construction, the quadrilaterals 퐴푆푀푊; 퐶푈푃푉; 퐵푅푁푇 are parallelograms. The middle of the diagonal 푊푆 falls on 퐴푀, so on the simedian 퐴퐾, and from Lemma 1 we get that 푊푆 is an anti-parallel to 퐵퐶. Since 푇푈 ∥ 퐵퐶, it follows that 푊푆 and 푇푈 are anti- parallels, therefore the points 푊, 푆, 푈, 푇 are concyclic (4).

3 Figure 2.

Analogously, we show that the points 푈, 푉, 푅, 푆 are concyclic (5). From 푊푆 and 퐵퐶 anti-parallels, 푈푉 and 퐴퐵 anti-parallels, we have that ∢푊푆퐴 ≡ ∢퐴퐵퐶 and ∢푉푈퐶 ≡ ∢퐴퐵퐶, therefore: ∢푊푆퐴 ≡ ∢푉푈퐶, and since 푉푊 ∥ 퐴퐶, it follows that the trapeze 푊푆푈푉 is isosceles, therefore the points 푊, 푆, 푈, 푉 are concyclic (6). The relations (4), (5), (6) drive to the concyclicality of the points 푅,푈,푉,푆,푊,푇, and the theorem is proved.

Remarks.

2. For any point 푀 found on the simedian 퐴퐴1 , by performing the constructions from hypothesis, we get a circumscribed circle of the 6 points of intersection of the parallels taken to the sides of triangle.

3. The Theorem 2 generalizes the Theorem 1 because we get the second in the case the parallels are taken to the sides through the simedian center 푘.

4 4. We get a circle built as in Theorem 2 from the first Lemoine circle by homothety of pole 푘 and of ratio 휆 ∈ ℝ.

5. The centers of Lemoine circles built as above belong to the line 푂퐾, where 푂 is the center of the circle circumscribed to the triangle 퐴퐵퐶.

Bibliography.

[1] Exercices de Géométrie, par F.G.M., Huitième édition, Paris VIe, Librairie Générale, 77, Rue Le Vaugirard.

[2] Ion Paatrascu, Florentin Smarandache: Variance on topics of Plane Geometry, Educational Publishing, Columbus, Ohio, 2013.

5