SEMI-EXCIRCLE OF QUADRILATERAL

Transcription

1 JP Journl of Mthemtil Sienes Volume 5, Issue &, 05, Pges - 05 Ishn Pulishing House This pper is ville online t SEMI-EXCIRCLE OF QUADRILATERAL MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI Deprtment of Mthemtis University of Riu Peknru, Riu Inonesi Astrt Suppose tht ABCD is qurilterl Then there will e 4 piees of semiexirle on the qurilterl In this pper, we will isuss how to etermine the length of the rii of the semi-exirle, ut previously we will lso e isusse how to etermine the length of 4 new sies forme from the extension of the sies on ABCD whih hs no prllel sies Introution On tringle lwys n e forme inirle n exirle [-, 7, 5-7], ut on ny qurilterl not neessrily n e forme inirle n exirle Anyone hve inirle ut o not hve exirle n there hs exirle ut o not hve inirle [,, 4, 7] Espeilly for exirle lrey isusse in [0-], ut in [0, ] only isusses out qurilterl hs n exirle For ny onvex qurilterl tully me y [] is not n exirle, euse only offen sie of the retngle n the other of two sies of extension, s in Figure 00 Mthemtis Sujet Clssifition: 5F0, 5M5, 5N5, 5N0 Keywors n phrses: semi-exirle of qurilterl, semi-gergonne, tngentil qurilterl, rii of exirle Reeive Novemer, 05

2 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI Figure In this pper, irle entere on the E, F, G n H (see Figure ) everything is lle semi-exirle of qurilterl ABCD We lle semi-exirle, for exmple irle entere in E, only offensive sie AB n extension of the CB n DA, s well s irle entere t F, G n H wheres R, R, R n R eh is the rius of the semi-exirle entere in E, F, G n H In [9-] is lle tngentil In [0, ] is not expliitly how long the fingers, ut n ssoition tht pplies is R R R R If ABCD is qurilterl tht there shoul e no prllel sie, then there will e two pirs of interseting sie (etils see Figure ) Suppose tht P AB I DC n Q AD I BC Then it nees to e isusse, how long of sie α BP, β CP, γ CQ n δ DQ If ABCD is yli qurilterl, to lulte the ρ AC length of eh hs een isusse on [7, 5, 7] for exmple α, with ρ is R

3 SEMI-EXCIRCLE OF QUADRILATERAL rius of the exirle wht if ADC n R rius is yli qurilterl The question is ABCD not yli qurilterl It is neessry to set the length of α, β, γ n δ n the length of rii of eh of the semi-exirle Figure Semi-Gergonne n Long of New Sie If on ny tringle, we lwys hve piees of outer of Gergonne point But in ny qurilterl hs not een le to onstrut outer of Gergonne point, euse it will not e possile to onstrut exirle for retngulr, exept ABCD hs n exirle, see [8, 0,, ] Beuse tht lwys n e onstrute is the offening one sie n the extension of the other two sies (on the onition tht there shoul e no prllel sies), so tht, eh piees of whih re inirle of BCP n CDQ were two others re exirle of APD n BQA But his fourth ws outsie of ABCD, so tht lines rwn to eh point of tngeny is lle the semi-

4 4 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI Gergonne of ABCD As for the four Gergonne points is like Figure Noteworthy tht I n I re the enter point of the inirle BCP n CDQ, while I n I eh of whih is enter point of exirle from APD n BQA While G e, G e, G e n e 4 G re Semi-Gergonne point on ABCD Figure Noteworthy tht impossile the inenter n e the sme s semi-gergonne point, exept the tringle tht forme is n equilterl tringle On ABCD onvex, if eh sie is extene, then there will e the sie tht meets in one point, then forms some new sies In Figure, there is no prllel of qurilterl sie Four sies re ifferent lengths etween eh other On Figure, n lwys e shown tht ADP ~ CBP, it follows tht α β β α

5 SEMI-EXCIRCLE OF QUADRILATERAL 5 So β α n α β whih result in α α, so tht α n β In nother prt, sine ABQ ~ CDQ, so tht So δ γ γ δ γ δ n δ γ

6 6 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI whih proue δ n γ Oviously the ove onitions only pply to the vlue of > n >, ut if otherwise if < n <, then the vlue of α, β, γ n δ will e negtive It is not possile, euse of the long sie is not possile negtive Therefore, the solute vlue is tken So the length of the sie of α, β, γ n δ eomes BP α, CP β, CQ γ, DQ δ

7 SEMI-EXCIRCLE OF QUADRILATERAL 7 Length of Rii of the Cirle Semi-exirle Length of rii of tngents irle qurilterl A onvex qurilterl of ABCD hs four exirle Eh of the exirle of onvex qurilterl n e lulte the length of rii Sine we hve een le to etermine the length of the sie of BP, CP, CQ n DQ, the length of rii of semi-exirle offensive sie of BC n e use the rii of inirle on BCP, tht is, R ( s α) tn BCP ( s β) tn CBP ( s ) tn BPC, where s ( α β) The vlue of R is lulte in full y using the vlue of,, n For exmple, if sustitute α n β to S, re otine s ( α β) () The sustitution of eqution () n the vlue of α to the vlue of will e otine R ove, it R ( s α) tn BCP

8 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI 8, tn BCP tn BCP In nother form n with similr reution ut y using the CBP will e otine ( ) CBP s R β tn CBP tn tn CBP But if using n, BPC then the result will e slightly ifferent, tht is, ( ) BCP s R γ tn BPC tn tn BPC Furthermore, to lulte the length of, R n e use inirle of, CDQ y using the formul ( ) CDQ s R γ tn ( ) DCQ s δ tn

9 SEMI-EXCIRCLE OF QUADRILATERAL 9 ( s ) tn CQD, where s ( γ δ) By using the vlue of,, n For exmple, if sustitute the vlue γ n δ to s, will e otine Thus otine s R ( s γ) tn CDQ tn tn CDQ If using DCQ, will e otine CDQ R ( s δ) tn DCQ tn tn DCQ Menwhile, if using CQD, will otin slightly ifferent results, nmely DCQ

10 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI 0 ( ) CQD s R tn CQD tn tn CQD While to lulting R n, R eh of whih is the sme s ounting exirle of APD n, ABQ sine ( ) s α β 4 n ( ) s δ γ so ( ) ( ) ( ) s β α 4 n ( ) ( ) ( ) s δ γ

11 SEMI-EXCIRCLE OF QUADRILATERAL Thus otine R tn APD n R tn BQA As si ove, the vlue of R, R, R n n the long sie of eqution ove for R is for se sie lengths of > > so if otherwise pplile, then the enomintor from the hnge into So generlly will e otine n hnge into R R R R R R R tn BQA, tn BCP, tn CBP, tn BPC, tn DCQ, tn CDQ, tn CQD, R tn APD

12 MASHADI, SRI GEMAWATI, HASRIATI AND HESY HERLINAWATI On the ove lultions, there re respetively formuls to lulte the vlue of R n R, viewe from ifferent ngle While the formul to lulte the vlue of n R n R eh only one formul This is euse the formul for lulting R use exirle formul, whih the exirle formul only unique While the formul to lulte R n R R erive from the inirle formul, whih inirle formul silly n e erive from three onepts of tht ngle [] A Gultierrez, Go Geometri, hl Referenes ongruenehtm sesse 4 August pm [] A Hess, On irle ontining the inenters of tngentil qurilterls, Forum Geom 4 (04), [] F Smne n I Ftrsu, The Geometry of Homologil Tringles, The Eutios Pulisher, In, Ohio, 0 [4] H Mowffq, A onition for irumsriptile qurilterl to e yli, Forum Geom 8 (008), 0-06 [5] H S M Coxeter n S L Greitzer, Geometry Revisite, the Mthemtil Assoition of Ameri, Wshington, D C, 967 [6] L Hoehn, A new formul onerning the igonls n sies of qurilterl, Forum Geom (0), - [7] Mshi, Pusngik Universits, Riu, 0 [8] Mshi, Sri Gemwti, Hsriti n Putri Jnurti, Some result on exirle of qurilterl, JP J Mth Si 4( & ) (05), 4-56 [9] Mihl Miulit, A new property of irumsrie qurilterl, Inter J Geom () (0), 6-64 [0] M Josefsson, On the inrius of tngentil qurilterl, Forum Geom 0 (00), 7-4 [] M Josefsson, More hrteriztions of tngentil qurilterl, Forum Geom (0), 65-8 [] M Josefsson, The re of ientri qurilterl, Forum Geom (0), 6-85

13 SEMI-EXCIRCLE OF QUADRILATERAL [] M Josefsson, Similr metri hrteriztions of tngentil n extngentil qurilterls, Forum Geom (0), 6-77 [4] N Minulete, Chrteriztions of tngentil qurilterl, Forum Geom 9 (009), -8 [5] M Ri, Z Krimn n V Kum, A onition tht tngentil qurilterl is lso horl one, Mth Commun (007), -5 [6] Pul Yiu, Introution to the Geometry of the Tringle, Flori Atlnti University, 0