SOME RESULT ON EXCIRCLE OF QUADRILATERAL

Transcription

1 JP Journal of Mathematical Sciences Volume 14 Issues 1 & 015 Pages Ishaan Publishing House This paper is available online at SOME ESULT ON EXILE OF QUDILTEL MSHDI SI GEMWTI HSITI and PUTI JNUTI Department of Mathematics University of iau Pekanbaru iau Indonesia bstract ny quadrilateral not necessarily have excircle in this paper will discuss necessary and sufficient condition that any quadrilateral having excircle. It also will set the various lengths of the sides are formed from the construction result of excircle. esides that we can also establish some other excircle and also be specified the length of radii and the relationship of the radii with the presence of excircle. 1. ackground In a triangle can always be constructed incircle and excircle [3 13] while in any quadrilateral may not necessarily be formed incenter and the excenter. Several authors discuss the incircle of a quadrilateral is [ ] the author only discusses about the comparative of radii side and diagonal of the quadrilateral. 010 Mathematics Subject lassification: 51F0 51N15 51N0. Keywords and phrases: excircle of quadrilateral gential quadrilateral radii of excircle. eceived September 015

2 4 MSHDI SI GEMWTI HSITI and PUTI JNUTI Figure 1.1 t [1] the author discusses the excircle as in Figure 1.1 which if we look then it is not an excircle because each circle just touches one side and extension of the other two sides. Furthermore [8 9] to construct an excircle (as Figure 1.) which gives the requirements that the quadrilateral D with sides a b D c and D d then D is gential quadrilateral if a c b d. Figure 1. However [8] also explained that if D convex quadrilateral where opposite sides and D intersect at J and the sides D and intersect at K (see Figure 1.3) then D is a gential quadrilateral if and only if either of J K DJ DK J K J K. In addition to the [9] also indicated that if the D convex quadrilateral with sides a b c d has an excircle if and only if a c b d as well as to construct

3 SOME ESULT ON EXILE OF QUDILTEL 43 various subtriangles so it obtained four pieces circumradii order to obtain the various relationships between the fourth circumradii. If we look at Figure 1. above of course the most import first is to prove that the bisector external bisector and D and opposite bisector are concurrence which then needs to be discussed how the radii of the excircle as well as a variety of side length resulting from the excircle constructing.. Theoretical asis In addition to calculating the radii of the circle gent can also be calculated disce of the center point to the one point of the outer angles of the triangle (external bisector). Josefsson in [7] lowers the formula in Lemma.1. Lemma.1. If O is the center of the incircle then will apply O ac s c s a with is a semiperimeter. Proof. See [10]. The constructed excircle on a convex quadrilateral produce concurrency of six bisector angle. To prove concurrency of six bisector angle can be done in the following manner. Note Figure.1. Note Figure.1 ΑΒΚ in Figure.1. Since of the circle centered at E offending side of K in point I the extension of K at the point H and the extension of at point F the

4 44 MSHDI SI GEMWTI HSITI and PUTI JNUTI circle is excircle of ΑΒΚ. y making a line bisector angle of each KF and KH then the third angle of bisector concurrence at the point E. Thus all three lines of bisector are E E and EK concurrence at point E. Note DJ. Since the circle centered at E offend the extension side of D at point H J extension at point F and DJ at point G the circle is excircle from DJ. y creating a line bisector angle of each JDK and DJF thirdline of the bisector angle namely E DE and EJ concurrence at point E. Points I and G is a point of gency of circle. y connecting the points and E are formed two triangles namely IE and EG so IE GE meaning that E is bisector line of KJ. Thus proven E E E DE EJ and EK concurrence. Theorem.. n outer circle gent quadrilateral with sides a b c and d has the length of radii Proof. See [10]. Theorem.3. Suppose the L D ρ a c L D. d b D d and also has a gential excircle. Then L D where γ is the number of opposite angles. D with a b D c and abcd( γ) Proof. onsider Figure.. Pull the line D so that there are two triangles that D and D at D applies: D a d ad. nd on the D the same thing is applicable that D b c bc.

5 SOME ESULT ON EXILE OF QUDILTEL 45 Thus obtained dditionally a d ad b Figure. c bc ( a d b c ) 4( ad bc ). (.1) L D can be written as L D L D L D 1 ad 1 bc 16L D 4( ad bc ) (.) add the equation (3.15) with (.) is obtained ( a d b c ) 16L D 4( ad bc ) 4( ad bc ) 4( a d b c ) 16abcd( γ) 8abcd ( ad bc) ( a d b c ) 16abcd( γ) [( ad bc) ( a d b c ) ][( ad bc) ( a d b c )] 16abcd ( γ) 16( s a) ( s b)( s c) ( s d ) 16abcd( γ)

6 46 ce MSHDI SI GEMWTI HSITI and PUTI JNUTI a b c d s so 16L D 16abcd 16abcd( γ) L D abcd( γ). 3. Sides and adii 3.1. Sides length The constructed of excircle on the quadrilateral produce various sides. For more details please see Figure 3.1. In Figure 3.1 there exists Figure 3.1 D with α. Suppose the length of HK e and FJ f. Since IK and HK are the gents of the point K IK HK e. Furthermore FJ and GJ are also gents from the point F so that GJ FJ f. Let H x. Then DK x d e. Furthermore ce of H and F are gents from point H F so J x a f. Since the DG and DH are also gents from point D DG DH so that DG x d which produces G x c d. Furthermore ce I G x c d the circle centered at point E is excircle from four triangles namely K DJ J and DK. Thus the semiperimeter ( s ) of each triangle is

7 SOME ESULT ON EXILE OF QUDILTEL 47 s 1 s K s 1 a b x c d e x d e d a b c d x. Since H F s K DJ x call s s DJ x and then we have s 3 s K x a (3.1) s 4 s DK y ug Lemma.1 in K and equation (3.1) x d. (3.) E K x x K E x a a( b x c d e) x ( b x c d e) x a E a( b x c d e) c d b e (3.3) and in a similar way would be obtained e EK ( x e) ( b x c d e) c d b e (3.4) f EJ d f ( x f ) ( x d f ) (3.5) ( x d ) DE d( x d f ) d f (3.6) f JE ( x a f ) ( x c d f ) c d f a (3.7)

8 48 MSHDI SI GEMWTI HSITI and PUTI JNUTI ( x a b) E b( x c d f ) (3.8) ( c d f a) e EK ( x d e) ( x c d e). (3.9) c e y substituting the equation (3.4) to (3.8) is obtained e( x e) ( b x c d e)( c e) e( x d e) ( x c d e) ( c d b e) d( x c) b( x d ) xd bcd cd d e. (3.10) ( d cd dx bx ) Furthermore to find the value of f substitution of equation (3.5) to the equation (3.7) is obtained f ( x f )( x d f ) ( c d f a) f ( x a f ) ( x c d f ) ( d f ) 3 f cx ax adx acd ad. (3.11) ad ac cx ax Note the EH. Since HE α it is obtained H o ( 90 α ) EH α H EH α. α In a similar way also be obtained e cd[ L D cot α ( a c) ( c b d )]. L D cot α( b d ) ad bcd (3.1) y substituting the equation (3.8) to the equation (3.7) is obtained f acd( a c d ) ( L D cot α( a c) ( a c) ( ad ac) ) (3.13)

9 SOME ESULT ON EXILE OF QUDILTEL 49 from x e and f and DH x d then DH L D cot α d a c I L D cot α a b. a c Furthermore by ug the principle of Pythagoras obtained E L D cot α a c L D a c L D (( cot α) 1) a c E L D cot α a a c L D a c [ L D cot α a( a c) ] ( a c) L D E DE L D cot α a b a c L D cot α d a c L D a c L D a c. 3.. adii of the other excircle Note Figure 3.. If incircle formed on J centered in O b and DK based in O c the circle is also an excircle. Thus the length of radii of excircle on D that offensive side of b and c can be solved by ug excircle theorem for the triangle.

10 50 MSHDI SI GEMWTI HSITI and PUTI JNUTI Figure 3. Note J. Suppose D θ. Then J 180 θ. So the length of the radii is symbolized by b is o b 1 o ( s J J ) ( 180 θ) ( J JF J ) ( 90 θ) o f cot θ b acd( a c d ) cot θ. L D cot α( a c) ( a c) ( ad ac) In a similar way would be obtained c cd[ L D cot α ( a c) ( c b d )] cot θ. L D cot α( b d ) ad bcd 3.3. elationship of radii of the other excircle The following are given characteristics of quadrilateral that has circle gent a second shape that associated with the circle gent first form. Theorem 3.4. Given a D has excircle if and only if. a b c d Proof. ( ) Suppose a D has excircle with long of radii a b c and d and also has a excircle in front of the point of. Will be shown that a b c d.

11 SOME ESULT ON EXILE OF QUDILTEL 51 Note Figure 3.3. a offensive in side a at the point of L offensive side c in point N and Figure 3.3 D the circle is a circle that is centered on the length of radii of the circle is a b c and the circle centered at O a O b O c and O d. O b offensive side b at the point M O a O c O d that offensive side of d at the point P. Suppose d each of which is the radii of The second form of excircle is the circles are in front of the point. Let the circle of offending extension of sides D and D respectively at point F I G and H. Thus the radii of the circle are ρ OH OI OG OF. Note OF ug trigonometric rules we obtained F ρ cot (3.14) OF o 180 o 90. So that OF ρ F F ρ. (3.15)

12 5 MSHDI SI GEMWTI HSITI and PUTI JNUTI y subtracting the equations (3.14) and (3.15) was obtained a ρcot. In a similar way would be obtained L a (3.16) L a. (3.17) y adding equations (3.16) and (3.17) was obtained L L a a a. (3.18) obtained y doing the same way to OI and OI and O b M and O b M b b. (3.19) y doing the same way to derived DE c G and E c G as well as O c N and DO c N y doing the same way to D c c. (3.0) OH and DOH and DO d P and O d P obtained D d d (3.1) and

13 SOME ESULT ON EXILE OF QUDILTEL 53 ρρ cot cot b a b a which will be equal to ρρ b a ρρ b a. b a ρρ (3.) eturn the same way will be obtained D D d c ρρ (3.3) ce D and. D So that D D

14 54 MSHDI SI GEMWTI HSITI and PUTI JNUTI by equations (3.) and (3.3) was obtained ρρ ab ρρ cd so ab cd. ( ) Suppose ab cd will be shown that D has excircle. Note Figure 3.4. Suppose the point of gency of the circle centered at the point O a O b O c and O d that points L M N and P. Figure 3.4 E a point is the center point formed by the intersection of each bisector J. E b point is the center point formed by the intersection of each bisector and and J. E c point is the center point formed by the intersection of each bisector and DK. nd E d point is the center point formed by the intersection of each bisector and DK. ρ a ρ b ρ c and the circle each centered on E a E b E c and a b c and d. So will apply ρ d are the lengths of the radii of E d and offend the extension side of ρ a ρb a b.

15 SOME ESULT ON EXILE OF QUDILTEL 55 s well as ρc ρd D D c d. nd ce it also applies D D then we obtain ρaρb a b ρcρd cd. Since ab cd it should ρ aρb ρcρd. From the definition of ρ a ρ b ρ c and ρ d then must ρ a ρb ρc ρd. Since the lengths of radii are same it must be the center point of the circle that is in other words D has excircle. E a Eb Ec Ed that are in front of point [1] Gultierrez Go Geometri hal 1 eferences ongruence.htm. acsessed 4 ugust pm []. Hess On a circle containing the incenters of gential quadrilaterals Forum Geom. 14 (014) [3] F. Samandace and I. Fatrascu The Geometry of Homological Triangles The Educatios Publisher Inc. Ohio 01. [4] H. Mowaffaq condition for a circumscriptible quadrilateral to be cyclic Forum Geom. 8 (008) [5] L. Hoehn new formula concerning the diagonals and sides of a quadrilateral Forum Geom. 11 (011) [6] Mihal Miculita new proferty of circumscribed quadrilatera Internat. J. Geom. 1() (01) [7] M. Josefsson On the inradius of a gential quadrilateral Forum Geom. 10 (010) [8] M. Josefsson More characterizations of gential quadrilateral Forum Geom. 11 (011) 65-8.

16 56 MSHDI SI GEMWTI HSITI and PUTI JNUTI [9] M. Josefsson The area of a bicentric quadrilateral Forum Geom. 11 (011) [10] M. Josefsson Similar metric characterizations of gential and exgential quadrilaterals Forum Geom. 1 (01) [11] N. Minculete haracterizations of a gential quadrilateral Forum Geom. 9 (009) [1] M. adic Z. Kariman and V. Kadum condition that a gential quadrilateral is also a chordal one Math. ommun. 1 (007) [13] Paul Yiu Introduction to the Geometry of the Triangle Florida tlantic University 01.