SOME RESULTS OF CONSTRUCTING SEMI GERGONNE POINT ON A NONCONVEX QUADRILATERAL

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1 Bulletin of Mathematics ISSN Printed: ; Online: Vol. 08, No , pp SOME RESULTS OF CONSTRUCTING SEMI GERGONNE POINT ON A NONCONVEX QUADRILATERAL Zukrianto, Mashadi, Sri Gemawati Abstract Suppose ABCD is a quadrilateral that is not convex. There are three pieces of semi-circular outer tangent and two pieces in a semi circle tangent of nonconvex quadrilateral. In this paper, it will be discussed how to calculate the length of the radius of the circle tangent quadrilateral that is not convex. Further discussion is how to calculate the length of the new sides that are formed by contructing the point of Semi Gergonne. 1. INTRODUCTION Two-dimensional figure of triangle is one of the scope of geometry that can always be constructed tangent circles inside and outside tangent circles [, 8, 13, 14]. If a line is drawn from the third vertex of the triangle to the point of tangency of the circle in incircle then the third line will intersect at one point concurrent, this is called Gergonne point inside the triangle and if there is a circle outside the triangle tangent excircle with the center point of tangency circle outside the triangle excenter, it also can be formed Gergonne point outside the triangle derived from the circle tangent [1, 9, 13]. But this does not apply to the quadrilateral because not all rectangles can be constructed in a tangent circle and outer circle tangent. Circle tangent in the quadrilateral [4, 6, 1] is a circle inside a convex quadrilateral and Received , Accepted Mathematics Subject Classification : 3B5, 51N15 Keywords and Phrases: quadrilateral, circle tangent, semi gergonne point 81

2 Zukrianto Some Results of Constructing Semi Gergonne Point... 8 the quadrilateral four sides of the offensive, while the outer circle tangent quadrilateral [4, 6, 1] is a circle that is offensive or lengthening the other side of a quadrilateral. Mashadi [10], have discussed about how to calculate the length of the radius of circle tangent to the convex quadrilateral, length of new side formed from the construction and how to construct of semi Gergonne point on a convex quadrilateral. Then Martin [7] also talked about the relationship between the length of the fingers outside the circle tangent to the convex quadrilateral. In the other article, Mashadi [11] also discusses the necessary and sufficient conditions in order that a quadrilateral has a tangent circle outside, calculate the length of the new sides and the length of the fingers and the relationship between the length of the fingers are formed. In addition to a convex quadrilateral, there is a quadrilateral having a single angle of more than 180. This is called quadrilateral is not convex [3]. In a quadrilateral is not convex can be constructed 3 tangent circles outer edges and in the semi circle tangent. In this paper, the author discusses the length of the fingers in a semi circle tangent length of the fingers and semi circle tangent outside as well as the relationship between the length of the radius of the semi circle tangent. It also will be determined length of new side formed from constructing the semi-circle tangent. To be more clear about quadrilateral is not convex which has a semi incircle and semi excircle tangent to tangent outside, consider Figure 1. In Figure 1, There are some tangent circle centered at the point I 0, I a, I b, I c and I d. Circle centered at the point I 0, I a and I b called semi-circle outside tangent quadrilateral is not convex ABCD, while a circle centered at point C, and D are called semi circle tangent in quadrilateral is not convex ABCD. Besides, there are some tangency circle radius outside is R 0, R a and R b and the radius of the incircle tangent is r c and r d. There is also a relationship between the length of the fingers formed, that is R a r d = R b r c.. SEMI GERGONNE POINT AND RADIUS LENGTH If any of the triangle can be constructed Gergonne point that is one piece Gergonne point inside the circle and 3 Gergonne point outside the circle [[1],[9],[13]], but the quadrilateral that is not convex can not be constructed, because the Gergonne point on quadrilateral that is not convex can not be constructed circle tangent. In quadrilateral is not convex only be constructed circle of offending one side and the extension of the other two sides that will formed three pieces circles tangent outside on ABP on the side of BP and AB and the circle tangent outside on AQD on the side of AD

3 Zukrianto Some Results of Constructing Semi Gergonne Point Figure 1 and two circles tangent in the ABP and AQD. Circle tangent formed on the rectangle is called semi circle tangent at quadrilateral is not convex ABCD. If a line is drawn from each vertex to the point of tangency in front of him, then the intersection of the three lines is called the point of the semi Gergonne at quadrilateral not convex ABCD. In Figure, there are six point of semi Gergonne can be constructed at quadrilateral is not convex ABCD that is Ge 1, Ge, Ge 3, Ge 4, Ge 5 dan Ge 6. A quadrilateral that is not convex ABCD has three outer edges tangent circles and two in the semi tangent circles. Each tangent circle can be calculated the length of his radius. Theorem.1 Suppose ABCD is a quadrilateral is not convex with the length of side are a, b, c and d then the length of the radius semi-circle tangent at the front corner of C is R 0 = L ABCD a c = L ABCD d b Proof.1 Let R o is the length of the radius semi-incircle tangent in front of the corner C as Figure 1. The area of the unconvex quadrilateral ABCD

4 Zukrianto Some Results of Constructing Semi Gergonne Point Figure can be formulated by L ABCD = L. ABO + L. ADO L. CBO L. DCO Let consider the ABO. Extension AB offensive circle at point S and OS is the radius symbolized R o then the height of ABO namely OS = R o, so the area of ABO is ABO = ASO BSO = 1 a + BSR o 1 BSR o = 1 a R o 1

5 Zukrianto Some Results of Constructing Semi Gergonne Point Let consider the ADO in Figure. Extension AD offensive circle at point R and OR is the radius symbolized R o then the height of ADO namely OR = R o, so the area of ADO is ADO = ARO DRO = 1 d + DRR o 1 DRR o = 1 d R o Let now consider the DCO in Figure. The side CD offensive circle at point U and UO is the radius symbolized R o then the height of DCO namely UO = R o, so the area of DCO is DCO = 1 c R o 3 L ABCD = 1.a.R d.R 0 1.b.R 0 1.c.R 0 for a + b = c + d is obtained L.ABCD = 1.a + d b c.r 0 R 0 = L.ABCD a c R 0 = L.ABCD a + d b c or R 0 = L.ABCD d b Theorem. Let ABCD is a quadrilateral is not convex with long side a,b,c dan d, then the area of quadrilateral is not convex ABCD is where γ = B+ D L.ABCD = sin γ Proof. In Figure.1, suppose the length of the side AB = a, BC = b, CD = c, AD = d and B + D = γ. Then connect point A and point C so that there are two triangles that ABC and ACD. At ABC cos rule applies, namely: AC = AB + BC.AB.BC. cos B AC = a + b.a.b. cos B 4

6 Zukrianto Some Results of Constructing Semi Gergonne Point And at ACD applies rules of sinus namely: AC = CD + AD.CD.AD. cos D AC = c + d.c.d. cos D 5 Based on the equation.1and., obtained a + b.a.b. cos B = c + d.c.d. cos D a + b c d =.a.b. cos B.c.d. cos D a + b c d =.a.b. cos B.c.d. cos D a + b c d = 4a.b. cos B c.d. cos D 6 Additionally L.ABCD can be written as L.ABCD = L. ABC + L. ACD = 1 ab sin B + 1 cd sin D L.ABCD = ab sin B + cd sin D 4L.ABCD =.ab sin B +.cd sin D 16L.ABCD =.ab sin B +.cd sin D 7 Add the equation.4 by.3, obtained 16L.ABCD + a + b c d =.ab sin B +.cd sin D + 4a.b. cos B c.d. cos D = 4ab sin B + cd sin D + 4a.b. cos B c.d. cos D = 4a b sin B + c d sin D + sin B sin D +4a b cos B + c d cos D cos B cos D = 4a b 1 cos B + c d 1 cos D + sin B sin D +4a b cos B + c d cos D cos B cos D = 4a b 4a b cos B + 4c d 4c d cos D + 8 sin B sin D +4a b cos B + 4c d cos D 8 cos B cos D = 4a b + 4c d + 8 sin B sin D 8 cos B cos D = ab + cd 8cos B cos D sin B sin D = ab + cd 8 cos B + D

7 Zukrianto Some Results of Constructing Semi Gergonne Point Because B + D = γ, so 16L.ABCD + a + b c d = ab + cd 8 cosγ = ab + cd 81 + cos γ = ab + cd 81 + cos γ 1 = ab + cd 16 cos γ 16L.ABCD = ab + cd 16 cos γ a + b c d = [ab + cd + a + b c d ][ab + cd a + b c d ] 16 cos γ = [a + ab + b c cd + d ][c cd + d a ab + b ] 16 cos γ = [a + b c + d ][c + d a + b ] 16 cos γ = [a + b + c da + b c + dc + d + a bc + d a + b] 16 cos γ Because a + b = c + d, so 16L.ABCD = [c + d + c dc + d c + da + b + a ba + b a + b] 16 cos γ = [cdab] 16 cos γ = cos γ = 1 cos γ = sin γ = sin γ By Theorem.1 and. is obtained that R 0 = a c sin γ or R 0 = d b sin γ Theorem.3 Let ABCD is a quadrilateral is not convex with long side a, b, c and d then the length of the radius semi-circle tangent outside on the side AD is R a = R 0cot A tan B cot B tan A

8 Zukrianto Some Results of Constructing Semi Gergonne Point Proof.3 Consider AI 0 S in Figure 1.1. obtained tan A = R 0 AS AS = R 0. cot A By using trigonometry rules 8 and consider BI 0 S at Figure 1.1, B tan tan 90 0 B cot B = R 0 BS = R 0 BS = R 0 BS BS = R 0 tan B 9 From equation.5 and.6 are obtained AB = AS BS a = R 0. cot A R 0 tan B a = R 0 cot A tan B 10 Look BI a V at Figure 1.1, tan B = R a BV BV = R a. cot B 11 Look AI a V at Figure 1.1, A tan tan 90 0 A cot A = R a AV = R a AV = R 0 BS

9 Zukrianto Some Results of Constructing Semi Gergonne Point AV = R a tan A 1 From equation.8 and.9 are obtained AB = BV AV a = R a. cot B R a tan A a = R a cot B tan A From equation.7 and.10 were obtained R a = R 0 cot A tan B cot B tan A Corollary.1 In the same way it will be obtained: 1. The length of the radius of the outer circle tangent that coincide the side AB and extend the side AD and BC on the non convex quadrilateral ABCD is R b = R 0 cot A tan D cot D tan A 15. The length of the radius of the outer circle tangent that coincide the sides AB, BC and the extension side CD at non convex quadrilateral ABCD is r c = R 0 cot B tan C cot B tan C The length of the radius of the semi circle in the coincide tangent AD, CD and the extension side BC pand the extension side ABCD is r d = R 0 cot D tan C cot D tan C 17 Furthermore, from the length of the fingers obtained, there is also a relationship between the length of the fingers results of construction with a length of the fingers as in the following theorem: Theorem.4 A quadrilateral is not convex ABCD with the long fingers of the semi circle tangent R a, R b, r c and r d, it will apply relationships R a.r d = R b.r c

10 Zukrianto Some Results of Constructing Semi Gergonne Point Proof.4 We know that is not a convex quadrilateral ABCD has long radius semi-circle tangent of R a, R b, r c and r d. By multiplying equation.11 to.14 is obtained R a.r d = R 0 cot A tan B cot B tan A. R 0 cot D tan C cot D tan C cot R a.r d = R0 A tan B cot D tan C cot B cot D R a.r d = R 0 R a.r d = R 0 cos A sin A cos B sin B tan A sin B cos B sin A cos A cos A cos B sin A sin B sin A cos B cos B cos A sin A sin B sin B cos A R a.r d = R0 cos A+B sin A cos B cos A+B sin B cos A cos D sin D cos D sin D tan C sin C cos C sin C cos C cos D cos C sin D sin C sin D cos C cos D cos C sin C sin D sin D cos C cos C+D cos D sin C cos C+D sin D cos C R a.r d = R 0 sin B cos A sin D cos C sin A cos B cos D sin C 18 Then, by multiplying the equation.1 to.13 is obtained R b.r c = R 0 cot A tan D cot D tan A. R 0 tan B cot C cot B tan C cot R b.r c = R0 A tan D tan B cot C cot D cot B R b.r c = R 0 cos A sin A cos D sin D tan A sin D cos D sin A cos A sin B cos B cos B sin B tan C cos C sin C sin C cos C

11 Zukrianto Some Results of Constructing Semi Gergonne Point R b.r c = R 0 cos A cos D sin A sin D sin A cos D cos A cos D sin A sin D sin D cos A R b.r c = R0 cos A+D sin A cos D cos A+D sin D cos A sin B sin C cos B cos C cos B sin C sin B cos C cos B cos C sin B cos C cos B+C cos B sin C cos B+C sin B cos C R b.r c = R 0 sin D cos A sin B cos C sin A cos D cos B sin C 19 By comparing the equation.15 to.16 is obtained R a.r d R b.r c = R0 sin B cos A sin D cos C sin A cos B cos D sin C R0 sin D cos A sin B cos C sin A cos D cos B sin C R a.r d R b.r c = 1 R a.r d = R b.r c

12 Zukrianto Some Results of Constructing Semi Gergonne Point Long Side New Construction Results In no convex quadrilateral ABCD there are several points of tangency circle centered at I 0, which are the point S, T, U and R See Figure 3.1. Suppose that BS = x, CT = z, and DR = y, then the length of the tangent lines can be determined by the following theorem: Theorem 3.5 Let ABCD squadrilateral is not convex, then the length of a tangent quadrilateral is not convex in the front corner of C on the extension of the side AB is x = a c. sin γ. cot A a Proof 3.5 See Figure 3.1. Figure 3.1 Suppose the length of the side AB, BC, CD and AD row is a,b,c dan

13 Zukrianto Some Results of Constructing Semi Gergonne Point d. By using the concept of trigonometry in Figure 3.1, it is obtained: tan A = I 0S AS tan A = R 0 AS AS = R 0. cot A AB + BS = a c. sin γ. cot A BS = a c. sin γ. cot A AB x = a c. sin γ. cot A a Corollary 3. In the same way will be obtained 1. The length of a tangent quadrilateral is not convex in the front corner of C on the side BC is z = a + b a c. sin γ. cot A. The length of a tangent quadrilateral is not convex in the front corner of C on the extension side AD is y = c a + b a c. sin γ. cot A Theorem 3.6 Let ABCD quadrilateral is not convex and Q is an intersection point between the lines AB din the extension of the line BC, then the length of the line AQ is AQ =. sin γ. cot A a c tan Q Proof 3.6 Notice Figure 3.1. Suppose the length of the side AB, BC, CD and AD row is a,b,c dan d. Then QAP = A, AQC = Q, and AP C = P. DBy using trigonometry comparison to the long side tangent earned AQ on ASI 0 is

14 Zukrianto Some Results of Constructing Semi Gergonne Point tan ASI 0 = tan QAP tan QAP = SI 0 AS tan A = R 0 AS AS = R 0. cot A and AQC tan SQI 0 = tan AQC tan = SI 0 QS tan 90 0 Q = R 0 QS cot Q = R 0 QS QS = R 0. tan Q so that AQ + QS = AS AQ + R 0. tan Q = R 0. cot A AQ = R 0. cot A R 0. tan Q AQ = R 0. cot A tan Q Because R 0 = a c. sin γ, then AQ =. sin γ. cot A a c tan Q Corollary 3.3 In the same way will be obtained:

15 Zukrianto Some Results of Constructing Semi Gergonne Point The length of the side AP which is the intersection of the line AD in the extension of the side BC is AP =. sin γ. cot A a c tan P. The length of the side CQ which is an extension of the line BC to the point Q is CQ =. sin γ. tan Q a c + cot A a + b 3. The length of the CP which is an extension of the line BC to the point P is CP =. sin γ. tan P a c + cot A a + b References [1] J.N. Boyd dan P.N. Raychowdhury, The gergonne point generalized through convex coordinates, Internat. J. Math. & Math.Sci. 1999, [] H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, the Mathematical Association of America, Washington, D. C., [3] F. Doddy, G. Agustinus dan H. Hedi, Geometri Bidang Datar, 1-3. [4] M. Hajja, A condition for a circumscriptible quadrilateral to be cyclic, Forum Geom , [5] M. Josefsson, More characterizations of tangential quadrilaterals, Forum Geom , [6] M. Josefsson, On the inradius of a tangential quadrilateral, Forum Geom , [7] M. Josefsson, Similar metric characterizations of tangential and extangential quadrilaterals, Forum Geom 1 01, [8] Mashadi, Geometry, Pusbangdik UR, Riau, 015. [9] Mashadi, Geometry Lanjut, Pusbangdik UR, Riau, 015.

16 Zukrianto Some Results of Constructing Semi Gergonne Point [10] Mashadi, S. Gemawati, Hasriati, dan H. Herlinawati, Semi excircle of quadrilateral, JP J. Math.Sci. 151 & 015, [11] Mashadi, S. Gemawati, Hasriati, dan P. Januarti, Some result on excircle of quadrilateral, JP J. Math.Sci & 015, [1] M. Nicusor, Characterizations of a tangential quadrilateral, Forum Geom 9 009, [13] F. Smarandache and I. Patrascu, The Geometry of Homological Triangles, The Educatios Publisher, Inc., Ohio, 01. [14] P. Yiu, Introduction to the Geometry of the Triangle, Florida Atlantic University, 01. Zukrianto: Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Riau zukrianto86@gmail.com Mashadi: Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Riau mashadi mat@gmail.com Sri Gemawati: Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Riau gemawati sri@gmail.com

SOME RESULT ON EXCIRCLE OF QUADRILATERAL

SOME RESULT ON EXCIRCLE OF QUADRILATERAL JP Journal of Mathematical Sciences Volume 14 Issues 1 & 015 Pages 41-56 015 Ishaan Publishing House This paper is available online at http://www.iphsci.com SOME ESULT ON EXILE OF QUDILTEL MSHDI SI GEMWTI