RELATIONS OF LEMOINE CIRCLE WITH A SYMMEDIAN POINT Ainul Wardiyah, Mashadi, Sri Gemawati

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 08, No. 01 (2016), pp. 69–79. http://jurnal.bull-math.org

RELATIONS OF LEMOINE CIRCLE WITH A

SYMMEDIAN POINT

_Abstract.

Ainul Wardiyah, Mashadi, Sri Gemawati

  Any triangle ABC have three symmedian lines that intersect at one

point K that called symmedian point, so triangle ABC can be partitioned into three

triangles, i.e. triangle ABK, triangle ACK and triangle BCK. From each of the

triangle can be constructed the circumcircle, for example with a center point R ,Q

and P respectively. If these points linked, so will form a triangle and has a centroid

point, say point M . In addition, each of the circumcircle triangle ABK, triangle

ACK and triangle BCK will intersect in six points on the sides and the extension

side triangle ABC which will be lie on one circle that known as third Lemoine circle.

If L is the center point of the third Lemoine circle, then in this article discuss that

K ,L and M are collinear.

  1. INTRODUCTION Triangle ABC have any special line symmedian [11] which intersect at one point called the point symmedian [1] which is denoted by K. So triangle ABC can be partitioned into three triangles, i.e. triangle BCK, triangle ACK and triangle ABK. In each triangle can be constructed circumcircle with the center point P , Q and R, respectively. If the points P , Q and R are linked by a line segment, it will form a triangle P QR. In addition, from constructing three circumcircle of triangle BCK, triangle ACK and triangle ABK, will form the intersection point of the circle with the sides and the extension side triangle ABC on six points. If the sixth point of

  Received 11-07-2016, Accepted 19-07-2016. 2010 Mathematics Subject Classification

: 97G40, 51A15

Key words and Phrases : Symmedian Line, Symmedian Point, Lemoine Circle, Centroid, Collinear.

  Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  intersection is connected, it will form a circle known as the third Lemoine circle [6]. In addition to the third Lemoine circle, there are the first Lemoine circle and second Lemoine circle [11, 12, 13, 14, 15]. The first Lemoine circle constructed by drawing three parallel lines to the sides of the triangle where all three lines through point symmedian K so it will intersect with the sides of the triangle at six points. Then the sixth point will be lie the one circle. While the second Lemoine circle constructed by drawing three anti parallel lines to the sides of the triangle are through the symmedian point so it will intersect in six points with a side of the triangle. Then the sixth point will be lie the one circle.

  This article will discuss the relationship of third Lemoine circle, sym- median point of triangle ABC and centroid point of triangle P QR. If M is the centroid point [8] of triangle P QR, and L is the center point of the third Lemoine circle while K is the symmedian point of triangle ABC so in this article will be proven that the three points K , L and M are collinear.

  2. SOME BASIC CONCEPTS Any triangle ABC has three median line [9] which, according to Ceva’s The- orem [10] will intersect at one point called the centroid point [8]. In addition to the median line, there are internal angle bisector also on a triangle, then if the A-median line reflected in the A-internal angle bisector will produce a line known as the symmedian line [11]. In any triangle, three symmedian lines will intersect at one point called the symmedian point [1] , and denoted by K.

  Definition 2.1 (Symmedian line). In a triangle ABC, the reflection of the A-median in the A-internal angle bisector is called the A-symmedian of triangle ABC. Similarly, we can define the B-symmedian and the C- symmedian of the triangle. Considering Figure 1, AM is median-A, AD and bisector angle-A. If AM is reflected in AD, so we have AX. The line AX is called symmedian-A.

  A circle can be constructed from a triangle. For example is circumcircle [7] with the point center is circumcenter, incircle triangle, and the tangent circle of triangle [6]. A circle that constructed from symmedian point is called Lemoine circle. L. Sammy and P. Cosmin divide the Lemoine circle in two forms, i.e. first Lemoine circle and second Lemoine circle. Grinberg

  Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  Figure 1: Symmedian Line and Symmedian Point. Lemoine circle and third Lemoine circle.

  Figure 2: First Lemoine Circle. Teorema 2.2

  (The first Lemoine Circle). Let K be the symmedian point of the triangle ABC and let x; y; z this time be the parallels drawn through K to BC, CA, and AB, respectively. Prove that the six points determined by x, y, z on the sides of ABC all lie on one circle.

  Proof . See [13]. Teorema 2.3

  (The Second Lemoine Circle). Let K be the symmedian point Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  of triangle ABC and let x, y, z be the antiparallels drawn through K to the lines BC, CA, and AB, respectively. Prove that the six points determined by x; y; z on the sides of ABC all lie on one circle. This is called the First Lemoine Circle of triangle ABC.

  Figure 3: Second Lemoine Circle. Proof . See [13]. Theorem 2.4

  (The Third Lemoine Circle) Let the circumcircle of triangle BLC meet the lines CA and AB at the points Ab and Ac (apart from C and B). Let the circumcircle of triangle CLA meet the lines AB and BC at the point Bc and Ba (apart from A and C). Let the circumcircle of triangle ALB meet the lines BC and CA at the points Ca and Cb (apart from B and A). Then, the six points Ab; Ac; Bc; Ba; Ca; Cb lie on one circle. This circle is a Tucker circle, and its midpoint M lies on the line U L and satisfies

  1 LM = LU (where the segments are directed). The radius of this circle is

  2

  1

  2

  2

  p9r + r , where r is the circumradius and r is the radius of the second

  1

  1

  2 Lemoine circle of triangle ABC.

  Proof . See [6]. Theorem 2.4

  (Centroid Theorem) If D , E and F respectively are the midpoints of the sides BC , CA and AB at a triangle ABC, then the line AD, BE and CF are concurent at point G and Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  Figure 4: Third Lemoine Circle.

  AG BG CG = = = 2

  GD GE GF Proof . See [10].

  3. RELATIONS OF LEMOINE CIRCLE WITH A SYMMEDIAN POINT

  In this article, the relationship of Lemoine circle that will be discussed is the relationship the third Lemoine circle with symmedian point of triangle ABC and centroid point of triangle P QR that formed by connecting the center point of the circumcircle of triangle AKB, triangle BKC and trian- gle AKC. If L the point center of third Lemoine circle, K symmedian point of triangle ABC, and M is centroid point P QR, will be shown that the K ,L and M are collinear. Theorem 3.1

  Let the symmedian point of triangle ABC is K, the center point of third Lemoine circle is L and M is the centroid of triangle P QR, Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  Figure 5: Median line and centroid point.

  Figure 6: Points K, L and M . Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  Proof K is the centroid point of triangle B a BB c , and B m is the point of intersection of the median line of of triangle B a BB c on the side B c B a and

  T is the point of intersection of the median-B line with triangle P QR , on the P R side, as shown in Figure 7.

  Figure 7: Triangle B BB with centroid K.

  a c

  Then by Theorem 2.4 it is shown that

  1 KB = BK

  m

  2 Then by connecting the points P and R with symmedian point and point B as shown in Figure 8. In order to obtain that the quadrilateral KRBP is a kite with two diagonal cut-off point at the point T , so

  1 T K = BK = KB m

  2 Then make a point S such that LT // SB as shown in Figure 9. Based on the theorem 2.4 shows that

  1 KL = KU,

  2 Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  Figure 8: Triangle B a BB c with centroid K.

  Figure 9: Point K, L, M , U and S. Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  maka

  1 KL/KS = KL/KU = .

  2 By dawing the line U N as shown in Figure9, we obtaine ∆KSB ∼ △KU N and by applying

  KU KS KL = =

  U N SB LT Since the point Q, K and N are collinear, then the point S, K and U are collinear too. By painting a line parallel to the U N through B m there will be a cut-off point on the segment of the U N in point H and by applying

  1 KH = KU

  2 With the enactment of this comparison, then the point K, H and U are collinear. In other words, L, K and H are collinear. Furthermore, M is the centroid of triangle P QR with J is median-Q, by drawing a line that through the U and parallel to the line KJ as in Figure 10.

  Figure 10: Triangle JV X. so

  KJ KM KM = = = 1/2 Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  KM = KU = KH With proven that KM = KH, it states that the point H = M . Thus concluded that the K, L and M are collinear.

  4. CONCLUSION The point center of third Lemoine circle of triangle ABC, symmedian point of ABC and the centroid point of triangle P QR are colinear.

  References

  [1] J. Casey, Analytic Geometry, 2nd edition,. Hodges and Figgis, Dublin, Ireland, 1893

  [2] J. L. Coolidge, A Treatise on the Geometry of the Circle and Sphere. , New York: Chelsea, 1971. [3] H. S. M. Coxeter, J.W. Introduction to Geometry, 2nd edition. John Wiley & Sons, Inc., New York, 1969. [4] H. S. M. Coxeter & S. L. Greitzer Geometry Revisited. Washington, DC: Math. Assoc. Amer, 1967. [5] M. Down Geometry. Addison Wesley Publishing Company.Inc, London, 1963. [6] D. Grinberg Ehramn’s Lemoine Circle. Mathematical Reflections, 3, 2007. [7] C. Kimberling. Central Points and Central Lines in the Plane of a Triangle. Math. Mag. 67, 163-187, 1994. [8] R. Lachlan An Elementary Treatise on Modern Pure Geometri. London: Macmillian, 1893. [9] E. Kohn Cliffs Quick Review Geometry. Penerbit Pakar Raya, Bandung, 2003. [10] Mashadi Geometri Lanjut. Pekanbaru: UNRI Press, 2015. [11] Mashadi Geometri. Edisi ke-2. Pekanbaru: UNRI Press, 2015.

  Ainul Wardiyah et al. – Relations of Lemoine Circle ...

  [12] N. Altshiller & Court. College Geometry (An Introduction to the modern geometry of the triangle and the circle), 2nd edition. New York:Dover Publications Inc, 2007. [13] R. Park. Symmedians. Tjusamo 2013-2014, January 13, 2014. [14] L.Sammy & P. Cosmin Let’s Talk About Symmedians, NC School of Science and Mathematics. Princeton University, USA, 1993.

  [15] Weisstein, Eric W. First Lemoine Circle. Resource.

  [http://mathworld.wolfram.com/FirstLemoineCircle.html]. [16] Weisstein, Eric W. Second Lemoine Circle. Resource.

  [http://mathworld.wolfram.com/FirstLemoineCircle.html].

  : Magister Student, Department of Mathematics, Faculty of

Mathematics and Natural Scinces Uneversity of Riau, Bina Widya Campus, Pekan-

baru 28293, Indonesia.

  E-mail: ai nul89@yahoo.co.id

  MASHADI : Analysis group, Department of Mathematics, Faculty of Mathematics

and Natural Scinces Uneversity of Riau, Bina Widya Campus, Pekanbaru 28293,

  Indonesia.

  E-mail: mashadi.mat@gmail.com

  : Analysis group, Department of Mathematics, Faculty of Math-

ematics and Natural Scinces Uneversity of Riau, Bina Widya Campus, Pekanbaru

28293, Indonesia.

  E-mail: gemawati.sri@gmail.com .