A NEW INEQUALITY AND IDENTITY

Size: px

Start display at page:

Download "A NEW INEQUALITY AND IDENTITY"

Brenda Harmon
5 years ago
Views:

1 Joural of Sciece ad Ar Year, No (), pp xx-xx, 0 A NEW INEQUALITY AND IDENTITY OIGINAL PAPE DAM VAN NHI Maucrip received: 00; Acceped paper: 00; Publihed olie: 000 Abrac I hi paper we iroduce he ew iequaliy ad ideiy called (M, N), ha Hayahi' iequaliy i oly a pecial cae The we will pree ome iereig applicaio Keyword: Hayahi' iequaliy, riagle, polygo 000 Mahemaical Subec Claificaio: 6D99, 97D50 INTODUCTION Suppoe give a riagle ABC of he legh of ide a, b, c Hayahi propoe a iequaliy: Wih ay poi M, we have amb MC bmc MA cma MB abc (ee [, ]) I hi paper we propoe a ew iequaliy which i a geeralizaio of he Hayahi' iequaliy, he we pree ome iereig applicaio i riagle Succefully, we have wo followig pricipal reul Theorem Le AA A be a polygo, be a ieger,, ad arbirary poi N, N,, N, M i euclidea plae we have he followig iequaliy MN AAMA i MAi i i A N We call hi iequaliy a ame he iequaliym, N If 0, we have Hayahi' iequaliy If,, ad A, B, C, N belog o he circle wih he ceer M we have he iequaliy aan bbn ccn 4S ABC School for gifed ude, Haoi Naioal Uiveriy of Educaio, Haoi, Vieam damvahi@yahoocom ISSN: Mahemaic Secio

2 6 Provig ome geomeric Dam Va Nhi Propoiio Aume ha he polygo AA A i icribed i he circle wih he ceer O ad radiu Taig poi N, N,, N ad M alo belogig o hi circle C Aumig ha he coo rdiae A co ;i,,,, ; he coordiae N co u ;iu,,,, ideiie ad he coordiae co ;i M u u The, we will have hee u co u u uu u i i i i i u i u i 0 u i i (iii) (iv) u i 0 if, i u u uu u i i if i i i (v) u u u i i i co ad 0 if i i i INEQUALITY AND IDENTITY Now we prove a i equaliy ha Hayahi' iequaliy i a p ecial cae Theorem Le AA A be a polygo, be a ieger,, ad arbirary poi N, N,, N, M i Euclidea plae we have he followig iequaliy wwwoaro Mahemaic Secio

3 Provig ome geomeric Dam Va Nhi 7 MN A N AAMA i MA i i i We call hi iequaliy a ame he iequaliy M, N If 0, we have Hayahi' iequaliy If,, ad A, B, C, N belog o he circle wih he ceer M we have he iequaliy aan bbn ccn 4S ABC Proof: Suppoe ha have affixe a, M ha affixe z ad N h affixe z h Uig he Lagrage ierpolaio formula, we have deducig A z z a z za a a za i i i i a z z z za a ai i i From hi, we deduce he geomeric iequaliy i ad MN AAMA i MA i i i A N ad he iequaliy M, N If 0 we have MN AN become he Hayahi' iequaliy for he polygo i MA AAMA i i i AN If, ad A, B, C, N belog o he circle wih he ceer M we have he iequaliy abc aa bbn ccn or aan bbn ccn 4 S ABC emar Deoe N a he ceer of circumcircle Applyig he iequaliy M, N or r [Euler] we deduce a b c ra b c Corollary Suppoe ha O, I ad G are repecively he ceer of circumcirle ad icircle of ABC Deoe he radii of circumcircle of he riagle GBC, GCA, GAB by,,, repecivly Le r a, r b, r c be he radii of circumcircle of he riagle IBC, ICA, IAB, ISSN: Mahemaic Secio

4 8 Provig ome geomeric Dam Va Nhi ad le ' ' ',, be he radii of circumcircle of he riagle OBC, OCA, OAB, repecively We have abc abc (ee []) (iii) ra rb rc where h a, h b, h c are he legh of aliude of ABC h h h r a b c (iv) ' ' ' x y z where ABC i o a obue riagle ad x, y, z are ha hb hc he diace from O o he ide, repecively Proof: Applyig he iequaliy abc abc M, N we obai aob OC boc OA coao B abc or Applyig he iequaliy, M N we obai agb GC bgc GA cga GB abc Sice S ABC abc abc abc agb GC 4S GBC 4 4 bgc G A 4, ad abc abc abc abc cga GB herefore abc or (iii) Applyig he iequaliy M, N we have aib IC bic IA cia IB abc Sice ra rabc ra rabc rb rabc rc rabc aib IC 4raS IBC rara 4, bic IA ad cia IB ha 4 ha h b h c ra rabc rb rabc rc rabc ra rb rc we have abc or ha hb hc ha hb hc r (iv) Applyig he iequaliy M, N we have aob OC boc OA coa OB abc Sice ' ' ' x abc ' x abc ' y abc aob OC 4S OBC xa 4, boc OA ha 4 ha hb ad ' ' ' ' z abc coaob we have y abc z abc abc h c ha hb hc or ' ' ' x y z h h h a b c Propoiio 4 Suppoe give a riagle ABC wih he legh of ide a, b, c repecively ad i he radiu of circumcircle of ABC Le' I, J a, Jb, Jc are he ceer of icircle ad ecribed circle of ABC, repecively The, wih ay poi M, we have wwwoaro Mahemaic Secio

5 Provig ome geomeric Dam Va Nhi 9 (iii) abcmi aai bbi cci MAMBMC MA MB MC MI abc bca cab abc MJ MJ MJ AJ AJ AJ BJ BJ BJ CJ CJ CJ a b c a b c a b c a b c (iv) MJa MJb MJb MJc MJ MJa AJa AJ AJ AJ AJ AJ MC bcma BJa BJb BJb BJc BJc BJa CJa CJb CJb CJc CJc CJa camb abmc c b b c c a Proof: Applyig he iequaliy M, N we have MI AI BI CI Sice bc b c a IA abc, ca c a b IB abc, ab a b c IC abc MI abc bca cab abc herefore (iii) Applyig he iequaliy M, N o,, we have he hree iequaliie MJa AJa BJa CJa MJb AJb BJb CJb MJc AJc BJc CJc O addig he hree iequaliie, we fid he iequaliy MJa MJb MJc AJa AJb AJc BJa BJb BJc CJa CJb CJc (iv) Applyig he iequaliy M, N o,, we have he hree iequaliie MJ MJ AJ AJ BJ BJ CJ CJ a b a b a b a b ISSN: Mahemaic Secio

6 0 Provig ome geomeric Dam Va Nhi MJb MJc AJb AJc BJb BJc CJb CJ MJc MJa AJc AJa BJc BJa CJc CJa c O addig he hree iequaliie, we fid he iequaliy MJa MJ b MJb MJ c MJc MJa AJa AJb AJb AJc AJc AJa BJa BJb BJb BJc BJc BJ MC bcma camb CJa CJb CJb CJc CJc CJa abmc a Corollary 5: Give a he riagle ABC of he legh of ide a, b, c ad i he radiu of circumcircle of ABC Deoe O, H he ceer of circumcircle ad he orhoceer o f ABC The, wih ay poi M, we have he iequaliy: abcmo MH aah bbh cch MC MA MB MC if M belog o he circle wih he ceer O ad he radiu, we obai he iequaliy abcmh a 4 a b 4 b c 4 c MC MA MB MC Proof: Applyig he iequaliy M, N o,, we have he iequaliy: M OMH AOAH BOBH COCH Thu, we obai he iequaliy AH a 4, BH b 4 ad a 4 a b 4 b c 4 c MA MB MC abcmo MH aah bbh cch Sice MC MA MB MC abcmh CH 4 c we obai MAMBMC Corollary 6: Suppoe give a riagle ABC wih he legh of ide a, b, c, repec ively Le I, G, H be he ceer of icircle, he ceroid ad he orhoceer of ABC The, wih ay poi M, we have he iequaliy abcmi aai bbi cci MAMBMC MA MB MC abcmg aag bbg ccg MAMBMC MA MB MC wwwoaro Mahemaic Secio

7 Provig ome geomeric Dam Va Nhi (iii) abcmh a 4 a b 4 b c 4 c MC MA MB MC Proof: Applyig he iequaliy M, N o,, N N N, we have MN AN BN CN abcmi aai bbi cci Therefore, we obai he iequaliy ad MAMBMC MA MB MC abcmg aag bbg ccg MAM B MC MA MB MC abcmh a4 a The we have ad If N H we have (iii): MC MA b4 b c4 c MB MC Example Suppoe give a riagle ABC of he legh of ide a, b, c repecively i he radiu of circumcircle; r, r, r are he radii of ecribed circle correpodece o verice A, B, C, repecively Le d a, d b, d c he diace from he ceer of circumcircle o h e ceer of ecribed circle The, wih ay poi D belog o he circumcircle of ABC we have he iequaliy: ddd a b c abc bc ca ab DJ DJ DJ a b c x b ca y cab z abc xyz abc r r r bc ca ab DJ a DJb DJc abc b ca cab abc xyz ab c Proof: We coider M O Sice bc a b c ca a b c ab a b c JaA, JaB, JaC bca bca bca bc b a c ca a b c ab b c a JbA, JbB, JbC cab cab cab bc c a b ca c b a ab a b c JA c, JB c, JC c abc abc abc ddd a b c a b c we obai abc bc ca ab DJ DJ DJ x bca y cab z abc xyz ab c herefore ISSN: Mahemaic Secio

8 Provig ome geomeric Dam Va Nhi Sice d a r, db r, d r herefore c r r r bc ca ab DJ a DJb DJc abc bca cab abc xyz ab c Propoiio 7 Le AA A be a polygo icribed i he circle wih he ceer O ad radiu The, wih ay poi N N i he plae AA A, we have he iequaliy AN i ONi i i i, i AN i If we obai AA i i i i, i AA If, ad a A A, a A A, a AA we obai he iequaliy ON aan aan aan 4S AA A i ON i Proof: Applyig he iequaliy M, N wih M O, we have he iequaliy AN i Le we obai AA AN ON i i i i AA i i, i i i i i, i Now, we illurae he advaage of he ideiy M, N by addreig everal impora problem of elemeary Geomery Firly, we ue he fucio i ad coi o creae he ideiy uder he form of rigoomery Wihou geeraliy, we ca aume ha he radiu of he circle C equal o Suppoe ha every poi A ha affixe a co ii, ad M ha affixe z couiiu ad every N h ha affixe zh couh iiuh From Lagrage ierpolaio formula, we have ON i zz a z z a za a a wwwoaro Mahemaic Secio

9 Provig ome geomeric Dam Va Nhi iu u i u uu u i i e i i e u u i i e i i e i i e iu iu i or We reduce all he facor i, e ad e, obai he relaio iu i i i u e u u i i i i uu u From hi relaio, we deduce wo ideiie below: i i co u u i u u u i i ad u u i u i 0 u i i From hi reul, we build he ideiie uder he form of rigoomery ad geomery for he iequaliy M, N a followig: Propoiio 8 Aume ha he polygo AA A i icribed i he circle wih he ceer O ad radiu Taig poi N N ad M alo belogig o hi circle C Aumig ha he coordiae A co ;i,,,,; he coordiae N cou;iu,,,, ad he coordiae M co u;iu The, we will have hee ideiie u u u i i u co u u i i i ISSN: Mahemaic Secio

10 4 Provig ome geomeric Dam Va Nhi (iii) u i u i 0 u i i u i 0 i if, (iv) u u u u u i i i i i if (v) co u u i i i u 0 i ad u i 0 if i ad emar 9 If he quadrilaeral ABCD i icribed i he circle we have by (iii) or ada cdc bdb DA DC DB DA DC DB bc ab ca Moreover, we have DA DB DC a DC DA DB c DB DC DAb Hece DA S DC S DB D [Feuerbach] DBC DAB DCA Propoiio 0 Suppoe he polygo AA A i icribed i he circle wih he radiu Taig poi N N ad M alo belogig o hi circle C Aumig ha he coordiae Aco ;i,,,, ; he coordiae N co u;iu,,,, ad he coordiae M co u;iu The, wih he proper choice of + or - we will have he ideiie MN A N u MA AA MA co, M, N wwwoaro Mahemaic Secio

11 Provig ome geomeric Dam Va Nhi 5 MA AN u i 0 A A (iii) MN A N co AA MA ad AN 0 AA Corollary Aume ha he poi wih he ceer O The, we have he ideiie AA A, M i order belog o he circle C r co MAr Ar MA A A MA r r r r r i MAr Ar 0 MA A A r r r r Proof: Thee ideiie follow from he ideiy M, N wih 0 Corollary Le he quadrilaeral ABCD be icribed i he circle C wih he ceer O Le a = BC, b = CA, c = AB The, we have wo ideiie: a co ODOA, co, co, b ODOB c ODOC abc DA DB DC DA DB DC ai ODOA, ci ODOC, bi ODOB, DA DC DB Proof: Thee ideiie follow from he ideiy M, N if, 0 CONJECTUE rue: Depie of o havig bee prove ye, hee followig reul are ill hoped o be Ope Problem Suppoe give a riagle ABC wih he legh of ide a, b, c repecively ad i he radiu of circumcircle of ABC Le' J a, J b, J c are he ceer of ecribed circle of ABC, repecively The, wih ay poi M, we have MJ MJ MJ AJ AJ AJ BJ BJ BJ CJ CJ CJ a b c a b c a b c a b c ISSN: Mahemaic Secio

12 6 Provig ome geomeric Dam Va Nhi MJ MJ MJ bc ca ab a b c abc bca cab abc MC MA MB MC Ope Problem Givig a riagle ABC wih he legh of ide a, b, c ad i he radiu of circumcribed circle; r, r, r are he radii of ecribed circle Le' d a, d b, d c he diace from he ceer of circumcribed circle o he ceer of ecribed circle The we alway have he iequaliy: ddd a b c abc bc ca ab x bca y cab z abc r r r bc ca ab abc b ca cab abc EFEENCES [] Adreecu, T, Adrica, D, Educaia Mahemaica, (), 9, 005 [] Hayahi, T, Thou Mah J, 4, 68, 9/4 [] Miriovic, DS, Pecaric, JE, Voleec, V, ece Advace i Geomeric Iequaliie, Kluwer Academic Publiher, Dordrech, Boo, Lodo 989 wwwoaro Mahemaic Secio

Ruled surfaces are one of the most important topics of differential geometry. The

Ruled surfaces are one of the most important topics of differential geometry. The CONSTANT ANGLE RULED SURFACES IN EUCLIDEAN SPACES Yuuf YAYLI Ere ZIPLAR Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey yayli@cieceaaraedur Deparme of Mahemaic Faculy of Sciece Uieriy