Nresun Universit Journl: Siene nd Tehnolog 08; 63 The Reltionship between the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle Yuporn Riholkrn Fult of Siene nd Tehnolog, Pibulsongkr Rjbht Universit, Phitsnulok, Thilnd 65000 orresponding uthor. E-il ddress: riholkrn@hoo.o.th bstrt The purpose of this reserh ws to stud the reltionship between the Nine-Point irle nd the irusribed irle of rhiedes tringle, nd the reltionship between the Nine-Point irle nd the Insribed irle of rhiedes tringle. The results were tht the rdius of the Nine-Point irle of rhiedes tringle is hlf the rdius of the irusribed irle of rhiedes tringle, nd the Nine-Point irle nd the Insribed irle of rhiedes tringle touh internll. Kewords: rhiedes tringle, Nine-Point irle, irusribed irle of tringle, Insribed irle of tringle Introdution The Swiss thetiin, Leonhrd Euler 707-783, disovered the Nine-Point irle of tringle, nd tht its iruferene psses through nine points, with the first three points being the idpoint of the tringle s sides, nd the three points originting t the perpendiulr line fro the verte to eet the opposite sides of the tringle, nd the other three points being the idpoint of the distne between the orthoenter nd the verte of the tringulr ngles. Dvis, 00. Krl Feuerbh 800-83, Gern thetiin, lso desribed this, in detil, Thus, this is lso lled the Feuerbh irle or Euler irle. L P L P M L M L 3 3 H P M 3 P M 3 L P 3 L P 3 M Figure the Nine-Point irle Figure ; H is the orthoenter of tringle, irle with its iruferene pssing through nine points whih re the idpoints of tringle s sides M, M, M 3, with the points originting t the perpendiulr line fro the verte to the opposite sides L, L, L 3 nd the idpoint of the orthoenter nd the verte of the tringles P, P, P 3 ll the Nine-Point irle of the tringle. The Greek thetiin, rhiedes 87-.., disovered tht the prboli segent of the prboli urve, nd the ord, link on two points on their prboli urve, then, fro lines touhing the prboli urve t the end of the ords long to the outside prboli segent, to for n rhiedes M 63
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Tringle. Thus, the bse of tringle is the prboli segent ord nd the other two sides re lines touhing the prboli urve t the end of the ords. Erbs, 000; Wolternn, 0 Figure. rhiedes Tringle Figure. the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle Figure.; n rhiedes Tringle fro the line eeting the prboli urve t point nd, side nd fro the lines tthing the prboli urve t nd nd eeting t point. ll of the Nine-Point irle, the irusribed irle nd the Insribed irle hve struturl reltionships with rhiedes Tringle Figure. thus, the is of this reserh were to desribe reltionship of the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle b using the nltil geoetr ethod to epnd the theti knowledge for developing other subjets. Methods nd Mterils This reserh used the following proedures; t first review bsi knowledge stud bout definition, propert, o-ordintes, oposition nd struture of the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle b using the nltil geoetr ethod, nd then find rdius of the Nine-Point irle nd the irusribed irle for desribe reltionship of the b using Eulidin geoetr nd retngulr nlti geoetr, solution of qudrti eqution, lulus nd rtio opre ethod, finll find distne fro enter of the Nine-Point irle to the Insribed irle for opre this with the differene of rdius of the Nine-Point irle nd rdius of the Insribed irle to desribe reltionship of the b using propert of irle nd Eulidin geoetr with trigonoetr proof. Results. si Knowledge. rhiedes Tringle Definition Let prboli segent is re tht s enlosed with prboli urve nd ord link on two points on their prboli urve, rhiedes Tringle is the tringle its bse is prboli segent ord nd other two sides re lines touh prboli urve t end of the ords.. rhiedes Tringle in Retngulr o-ordinte Sste 6
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Deterine rhiedes Tringle in Retngulr o-ordinte sste is result of line +, where nd re rel nuber nd not equl to 0 siultneousl eet prboli urve, where is rel nuber nd not equl to 0 t point nd. ord is bse of this tringle, side nd originted fro two tngent line of prboli urve t nd, point of intersetion is. + 3. 3. Figure 3 rhiedes Tringle in Retngulr o-ordinte Sste Mnoosilp 0 nd Riholkrn 07 studied to o-ordinte verte of rhiedes Tringle b solving eqution for intersetion point of line nd prboli urve nd intersetion point of line nd. then, the found;, -,,.3 Orthoenter of rhiedes Tringle Definition The Orthoenter of tringle is the intersetion point originting fro perpendiulr line fro the verte eet the opposite tringle sides. Dunh, 998 rhiedes Tringle is result of line +, eet prboli urve, t point nd. Let L, L nd L 3 is the point originted the perpendiulr line fro the verte eet the side, nd respetivel. H Orthoenter is intersetion point of line L, L nd L 3. L L H L 3 Figure Orthoenter of rhiedes Tringle + 65
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Finding o-ordinte of Orthoenter of rhiedes Tringle Figure ; Line L perpendiulr to, fro the derivtive of nd substitute with thus, the slop of line, therefore, slop of line L - o-ordinte of point t, thus, liner eqution L is --- Fro line L 3 perpendiulr to the slop of line, therefore, slop of line L 3 - o-ordinte of point t,, thus, liner eqution L is - --- Finding o-ordinte of point H fro intersetion line L 3 nd L onsider nd result fro -: 66
Nresun Universit Journl: Siene nd Tehnolog 08; 63 substitute in then, the o-ordinte of orthoenter of rhiedes Tringle t H,. The Nine-Point irle nd the irusribed irle of rhiedes Tringle Definition 3 the Nine-Point irle of rhiedes tringle is the irle tht its iruferene pss through the three idpoints of tringle sides, then three points originted perpendiulr line fro verte eet opposite tringle sides nd three idpoint of distne between orthoenter nd verte of rhiedes Tringle.. Rdius of the Nine-Point irle of rhiedes Tringle Deterine the iruferene of the Nine-Point irle of rhiedes Tringle pss through point L, P, L, M, P 3, M 3, L 3, P nd M respetivel, where M, M nd M 3 is the idpoint of side, nd, point L, L nd L 3 originted the perpendiulr line fro verte eet side, nd, point H is orthoenter nd point P, P, nd P 3 is the idpoint of the line H, H nd H respetivel. M L P 3 H P L M 3 L 3 M P + M L N P + 5. 5. Figure 5 Rdius nd re of the Nine-Point irle of rhiedes Tringle 67
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Figure 5.; Point M, L nd P re on the iruferene nd M L P is right ngle, thus, M L P is n ngle in seiirle nd line M P is the dieter of the Nine-Point irle. Finding o-ordinte of Point M idpoint of side M is idpoint of side, thus, o-ordinte of M is then, o-ordinte of the idpoint of side t M, Finding o-ordinte of Point P idpoint of line H P is the idpoint of the distne fro, - to H,, thus, the o-ordinte of point P is, 3, 3 then, o-ordinte of idpoint of line H t P, Finding the o-ordinte of enter of the Nine-Point irle of rhiedes Tringle Let point N is the enter of the Nine-Point irle of rhiedes Tringle idpoint of line M P, thus, o-ordinte of point N is 3, 5 8, 8 5 8 then, the o-ordinte of enter of the Nine-Point irle t P, Finding the length of Rdius of the Nine-Point irle of rhiedes Tringle Let r n P N is the rdius of the Nine-Point irle of rhiedes Tringle 8 68
Nresun Universit Journl: Siene nd Tehnolog 08; 63 r n P N 5 3 8 8 5 6 8 8 8 6 8 6 8 6 6 6 5 8 86 6 then, the length of Rdius of the Nine-Point irle of rhiedes Tringle is equl to 6 5 8 86 6 6 unit. Rdius of the irusribed irle of rhiedes Tringle Definition the irusribed irle of the tringle is the irle pssing through the three verties of the tringle. Yiu, 00 Definition 5 the iruenter of tringle is the intersetion point of the perpendiulr bisetors of the three sides of the tringle. Dunh, 998 Deterine rhiedes Tringle is result of line +, eet the prboli urve, t point nd. Let M, M nd M 3 is the idpoint of side, nd respetivel. O is the intersetion point of the perpendiulr bisetors of the three sides of rhiedes Tringle. + 6. 6. M O M M 3 Figure 6 Rdius nd re of the irusribed irle of rhiedes Tringle + 69
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Finding o-ordinte of iruenter of rhiedes Tringle o-ordinte of idpoint of side is M, nd slop of, fro the M O perpendiulr to, therefore, slop of line M O thus, liner eqution of M O is ------ 3 Point, - nd, re the end point of side nd M 3 is the idpoint of side, thus, the o-ordinte of point M 3 is,, then, o-ordinte of idpoint of side t M 3, the slop of side the line M 3 O perpendiulr to, therefore, slop of line M 3 O The liner eqution of line M 3 O is - ----- 70
Nresun Universit Journl: Siene nd Tehnolog 08; 63 7 Point O is the iruenter of tringle nd lso the intersetion point of line M O nd M 3 O, thus, fro 3 nd the o-ordinte of point O is substitute in 3 then, the o-ordinte of iruenter of rhiedes Ttringle t M 3, Finding length of Rdius of the irusribed irle of rhiedes Tringle Let R O is rdius of the irusribed irle of rhiedes tringle R O 3 6 6 6 8 6 8 6 6 6 6 8 8 5
Nresun Universit Journl: Siene nd Tehnolog 08; 63 then, length of Rdius of the irusribed irle of rhiedes Tringle is equl to 5 8 86 6 6 unit 3. The Reltionship between the Rdius of the Nine-Point irle nd the irusribed irle of rhiedes Tringle Figure 7 Reltionship between the Nine-Point irle nd the irusribed irle of rhiedes tringle Fro figure 7; let r n is rdius of the Nine-Point irle of rhiedes tringle R is rdius of the irusribed irle of rhiedes tringle the rtio of the rdius of the Nine-Point irle of rhiedes tringle nd rdius of the irusribed irle of rhiedes tringle is r n R + 5 5 8 8 then, the rdius of the Nine-Point irle of rhiedes Tringle is hlf of the rdius of the irusribed irle of rhiedes tringle Rerk Let r nd R is the rdius of the Nine-Point irle nd the rdius of the irusribed irle of rhiedes Tringle respetivel. the rdius of the Nine-Point irle of rhiedes Tringle is hlf of the rdius of the irusribed irle of rhiedes Tringle, so tht, we n s R r. The Nine-Point irle nd the Insribed irle of rhiedes Tringle Definition 6 the Insribed irle of tringle is the irle tht its iruferene touhes three sides of tringle. The Inenter of tringle is intersetion point of the three ngles bisetors of tringle. 86 6 86 6 6 6 7
Nresun Universit Journl: Siene nd Tehnolog 08; 63. Distne fro iruenter to Inenter of rhiedes Tringle Deterine point O nd I is the iruenter nd inenter of rhiedes Tringle respetivel, point Q, Q nd Q 3 is the point of tngen of the iruferene of the Insribed irle, line IQ, IQ nd IQ 3 re the rdius of the Insribed irle nd line OI is the distne fro the iruenter to the inenter of the rhiedes Tringle. L + + Q I I U O Q Q 3 Q U L 8. 8. Figure 8 Reltionship between the Nine-Point irle nd the Insribed irle of rhiedes tringle Finding distne fro iruenter to Inenter of rhiedes Tringle onsider Figure 8.; let R is rdius of the irusribed irle of rhiedes Tringle r is rdius of the Insribed irle of rhiedes Tringle d is distne fro point O to I Drw line fro point through I eet the iruferene of the irusribed irle t point L, us, line L is bisetor of, so tht L is idpoint of urve Drw line fro point L through O eet the iruferene of the irusribed irle t point L, thus, line L L is dieter of the irusribed irle Q is point of tngen, thus, IQ is the perpendiulr line to side t point Q nd line IQ r is rdius of the Insribed irle Tringle Q I nd L L re siilr tringle beuse Q I L L both re right ngle Q I L L both ngle on urve L re equl IQ L L if two pirs of two tringles re equl, so is the third ngle Fro siilr tringle Q I nd L L, the rtio IQ : L I: L L IQ L L I L Rr I L ----------- 5 Drw line I, let nd onsider tringle I, IL nd tringle L I, IL, thus, IL IL, IL is isoseles tringle nd side L IL substitute IL in 5 therefore, Rr I IL 73
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Drw line fro point O through I eet the iruferene of the irusribed irle t point U nd drw line fro point I through O eet the iruferene of the irusribed irle t point U, thus, line U U is dieter of the irusribed irle nd U U R Fro U I U I I IL Rr, thus R + d R - d Rr R - d Rr d R R - r OI R - Rr ------------- 6 then, the distne fro the iruenter to the inenter of rhiedes Tringle is in the for OI R - Rr. Distne fro Inenter to Orthoenter of rhiedes Tringle Deterine point I nd H is the inenter nd orthoenter of the rhiedes Tringle respetivel, line IH is the distne fro the inenter to the the orthoenter of rhiedes Tringle. Figure 9 inenter nd orthoenter of rhiedes Tringle Finding the distne fro the Inenter to the Orthoenter of the the rhiedes Tringle Let R is rdius of the irusribed irle of rhiedes Tringle r is rdius of the Insribed irle of rhiedes Tringle, nd is, nd respetivel Fro H 90 nd I, thus, HI Line H Ros nd I onsider tringle HI, fro the Lw of osine, IH H 90 I L H I R sin sin HI os + R os 6R sin sin R os R sin sin os R os 6R sin sin 6R os sin sin os os sin sin 7
Nresun Universit Journl: Siene nd Tehnolog 08; 63 R os sin sin R os sin sin os os sin sin R os sin sin R os sin os sin os os sin sin R os 8sin sin os os sin sin R os 8sin sin sin os sin sin R 8sin sin sin osos sin sin R 8sin sin sin osos 80 sin sin R 8sin sin sin os sin sin Rsin sin sin R os os os r - R os os os IH r - R os os os ----------- 7 then, the distne fro the inenter to the orthoenter of rhiedes the Tringle is in the for IH r - R os os os.3 Distne fro iruenter to Orthoenter of rhiedes Tringle Deterine point O nd H is the the iruenter nd the orthoenter of rhiedes Tringle respetivel, O is rdius of the irusribed irle, OH is the distne fro the iruenter to the orthoenter of the rhiedes Tringle. L M H O + Figure 0 the iruenter nd orthoenter of rhiedes Tringle 75
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Deterine point O nd H is the iruenter nd the orthoenter of the rhiedes tringle respetivel, O is the rdius of the irusribed irle, OH is the distne fro the iruenter to the orthoenter of the Tringle. Let R is rdius of the irusribed irle of rhiedes tringle r is rdius of the Insribed irle of rhiedes tringle, nd is, nd respetivel Fro H 90 nd O 90, thus, HO 90 90 - onsider tringle HO, fro Lw of osine, os - H O OH HO R os R OH R os R R R osos os R OH R os OH R os + R - R os os - R + R os os - os - R - R os os + - os - R - R os os os R - 8R os os os ------------ 8 then, the distne fro the iruenter to the orthoenter of the rhiedes Tringle is in the for OH R - 8R os os os. Distne fro the enter of the Nine-Point irle to the Inenter of the rhiedes Tringle Definition 7 Two irles re touh internll; iff the differene of the length of their rdius is equl to distne between their enters. Deterine point O, I, N nd H is the iruenter, inenter, nd enter of the Nine-Point irle nd the orthoenter of the rhiedes Tringle respetivel. R os R OH 76
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Q H I O N P + H I N. O. Figure the Nine-Point irle nd the Insribed irle of rhiedes Tringle Let O R is rdius of the irusribed irle of rhiedes Tringle NP R is rdius of the Nine-Point irle of rhiedes Tringle IQ r is rdius of the Insribed irle of rhiedes Tringle IN is distne fro enter of the Nine-Point irle to inenter of rhiedes Tringle onsider figure.; beuse of the enter of the Nine-Point irle, iruenter nd orthoenter of the the tringle re olliner, nd enter of the Nine-Point irle is the idpoint of the iruenter to the orthoenter, Yiu, 998, therefore, the line is the edin of tringle HIO Deterine the length of line HO, IO nd HI is equl, b nd respetivel, so tht IN b IN Fro 6, 7 nd 8 IN IN OI IH OH R Rr r R os os os R R Rr r 8R os os os R R Rr r R r R r R r R r 8R 8R os os os os os os 77
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Then. the Distne fro enter of the Nine-Point irle to the Inenter of the rhiedes Tringle is equl to the differene of the length of their rdius. 5. The Reltionship between the Nine-Point irle nd the Insribed irle of rhiedes Tringle Fro topi.; the differene of the length of the rdius of the Nine-Point irle nd the Insribed irle is equl to the distne between their enters, b definition 7 the result is two irles re touh internll. Then, the reltionship of the two irles is the Nine-Point irle nd the Insribed irle of rhiedes Tringle re touh internll. Rerk This reserh fous on the se prboli urve onl, beuse Even though the urve ple on nother where in retngulr o-ordinte sste, we n ertinl fort b sliding the is, djust or hnging the vrible to the for. Disussion This reserh point out studing bout the reltionship of the Nine-Point irle, the irusribed irle nd the Insribed irle of rhiedes Tringle b using nltil geoetr nd trigonoetr proof. the result reveled tht; the rdius of the Nine-Point irle of n rhiedes Tringle is hlf of the rdius of the irusribed irle of the rhiedes Tringle; this knowledge disover is in ord with the reserh of ook 99, ourt 980 nd lso Hung 0, their results showed tht; the rdius of the Nine-Point irle is hlf of rdius of the irusribed irle of tringle. Other result of this reserh is the Nine-Point irle nd the Insribed irle of rhiedes Tringle re touh internll; knowledge disover is in ord with the reserh result b using projetive geoetr proof reserhes of Krishn 06, reserh result b verte hnging to the for ople nuber of Yiu 998 nd lso reserh result of Dekov 009 whih found tht; the Nine-Point irle nd the Insribed irle of rhiedes Tringle re touh internll, thus, this reserh is in line with deiins nd ording to the hpothesis. onlusion nd Suggestion The result of this reserh showed tht; the rdius of the Nine-Point irle of n rhiedes Tringle is hlf of the rdius of the irusribed irle of rhiedes Tringle nd is the Nine-Point irle nd the Insribed irle of rhiedes Tringle re touh internll, this onlusion n ondue towrds knowledge etending other topi bout rhiedes Tringle nd the enter, rdius or re of the Nine-Point irle of rhiedes Tringle. In ddition; none interested ought to stud the propert or reltionship of Spieker irle, Pedl irle, Ngel-Point or Gergonne-Point, et. knowledgeent We re Thnkful to the Reserh nd Developent Institute, Pibulsongkr Rjbht Universit, for their finnil support, nd ll Prtiipnts, who ontributed to the oplishent of this reserh. 78
Nresun Universit Journl: Siene nd Tehnolog 08; 63 Referenes ook, N. M. 99. Tringle nd ITS irles. Knss: Knss Stte griulturl ollege. ourt, N.. 980. ollege Geoetr nd ed.. New York: Dover Publition, In. Dvis, T. 00. Four Points on irle. Retrieved M 9, 05, fro http://www.geoeter.org/ thirles Dekov, D. 009. oputer-generted Mthetis: the Feuerbh Point. Journl of oputer-generted Euliden Geoetr, 3, -6. Dunh, W. 998. Euler the Mster of Us ll. Wshington, D: the Mthetil ssoition of eri. Erbs, K.. 000. n Eplntor pproh to rhiedes's Qudrture of the Prbol. Retrieved fro http://jwilson.oe.ug.edu/emt668/emt6680.f99/erbs/et6690/ess/ ess.htl Hung, F. 0. Rdius of 9-Points irle Hlf of irurdius. Retrieved fro http://www.hkedit. net/ihouse/fh7878/ Krishn, D. N. V. 06. Yet nother New Proof of Feuerbh s theore. Globl Journl of Siene Frontier Reserh F,, 9-6. Mnoosilp, P. 0.the reltionship between the entroid of rhiedes Prboli Segent nd the entroid of rhiedes Tringle. Srinkhrinwirot Siene Journl, 30, 5-65. Riholkrn, Y. 07. Reltionship between re of rhiedes Prboli Segent nd re of rhiedes Tringle. Nresun Universit Journl Siene nd Tehnolog, 5, 58-67. Wolternn, M. 0. rhiedes Squring of Prbol. Retrieved fro www.wshjeff.edu/ users/ wolternn/dorrie/56.pdf Yiu, P. 998. Euliden Geoetr. Florid: Florid tlnti Universit. Yiu, P. 00. Introdution to the Geoetr of the Tringle. Florid: Florid tlnti Universit. 79