ON AN INEQUALITY FOR THE MEDIANS OF A TRIANGLE

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1 Journl of Siene nd Arts Yer, No. (9), pp. 7-6, OIGINAL PAPE ON AN INEQUALITY FO THE MEDIANS OF A TIANGLE JIAN LIU Mnusript reeived:.5.; Aepted pper:.5.; Pulished online: Astrt. In this pper, we give two new sipler proofs of shrp inequlity for the edins of tringle. We lso estlish two new inequlities y using this shrp inequlity. Soe relted onjetures heed y the oputer re put forwrd, whih inlude two onjetures relted to the fous Erdös-Mordell inequlity. Keywords: tringle, edin, inequlity, Erdös-Mordell inequlity Mthetis Sujet Clssifition: 5M6.. INTODUCTION In, X.G. Chu nd X.Z.Yng [] estlished the following geoetri inequlity: Let ABC e tringle with edins,,, irurdius, inrdius r nd seiperieter s. Then the following inequlity holds: 4s 6r 5r, (.) with equlity if nd only if ABC is equilterl. This is strong inequlity nd hs soe pplitions (see e.g. [], []). In y reent pper [], I hve shown tht the ointionl oeffiients in (.) is the est possile. In ft, y Theore in [] it is esy to prove the following onlusion: For ll inequlities in the for s r r, (.) inequlity (.) is the est possile, where,, re onstnts nd stisfy = 8. On the other hnd, it is interesting tht there exists the following shrp inequlity (.) whih is stronger thn (.): Theore. In ny tringle ABC with sides,,, edins,,, inrdius r, nd irurdius, the following inequlity holds: r, (.) Est Chin Jiotong University,, Nnhng City, Jingxi Provine, Chin. E-il: hin99jin@6.o. ISSN: Mthetis Setion

2 8 On n inequlity for Jin Liu with equlity if nd only if tringle ABC is equilterl. er.. If ΔABC ight e degenerte tringle, then the equlity in (.) would lso rrive t the se when A =, B = C =. This ft shows inequlity (.) is shrp. H.Y.Yin first posed n equivlent for of (.) when the inequlity (.) just hd een set up (see [], [4]). Until reently, (.) hs een proved y the uthor in []. However, this proof is very oplited. The uthor used le in [], i.e. the inequlity: s 4s 4, (.4) with equlity if nd only if =. In this pper, we give two sipler proofs of Theore, oth of whih do not depend on (.4). We lso give two pplitions of Theore. One of the is eutiful liner inequlity involving the edins nd the ltitudes of tringle. Another result is out the ute-ngled tringle. In the lst setion, we will propose soe relted onjetures.. NEW POOFS OF THEOEM In this setion, we will give two proofs of Theore. To siplify tter, we denote yli sus nd yli produts y, respetively. Proof : (The ethod of r s) By Cuhy inequlity, we hve i.e.,. (.) Therefore, to prove inequlity (.) we need to prove tht Using the nown forul 4 (.) is equivlent to r. (.), it is esily nown tht inequlity r. (.) Sine Mthetis Setion

3 On n inequlity for Jin Liu 9 hene (.) is equivlent to Thus, we hve to prove tht 4 4 r 5. X r 5 4. (.4) Using the following nown identities (see e.g. [5]):, we otin 4rs, (.5) s 8rs r, (.6) 4 rs r 4 r 4 rs r 4 r 4 s r, (.7) 4 4 s 4 r, (.8) ( ) ( ) (4 8 ) (4 ) 6 4 s r s r r r s r r r, (.9) X 4rX, (.) where X s rr s r r r r s 4 r r Oviously, the proof of X is hnged to X. If we put (.) G T 4 s 4 4r r then it is esy to verify the following identity: where X X s, r r s r4 r, GT X s, (.) s 6 rrs 4 8 r 84 r 56r 4r r r 4 r r. s ISSN: Mthetis Setion

4 On n inequlity for Jin Liu By identity (.), Gerretsen inequlity G nd the fundentl inequlity T of tringles (see [5], [6]), to prove X it reins to prove tht s X (.) 4 4 Let K 46 rrs 4 8 r 84 r 56r 4r, then it is esy to show tht K y e non-negtive nd lso e negtive y giving exples. So we n divide the proof of X s into the following two ses, i.e. K nd K. Cse. Assuing K. In this se, ording to the property of prols nd the Gerretsen inequlities: 6r 5r s 4 4r r, (.4) s X is stritly inresing on the intervl 6r 5r, 4 4r r to prove tht X 6r 5r, ut X 6r5r r 6r6r5r 4 8 r84 r 56r 4r 6r5r 4 4 rr r r r r r r r 4 = So we only need The ltter inequlity follows fro Euler inequlity under the first se. r. Hene s X is proved Cse. Assuing. For this se, it is esy to now tht X s is deresing on 6r 5r, 4 4r r. Thus we only need to show X 4 4r r. Siple oputtions give X K r r 44 4 r 7 r 4r r r Therefore X s is vlid under the seond se. Coing with the rguents of the two ses, s X holds for ll tringle ABC. Therefore, (.4), (.), (.) nd (.) re ll proved. Fro the dedutions ove, it is ler tht the equlity in (.) holds only when ΔABC is equilterl. The proof of Theore is oplete. Proof : (The ethod of the Differene Sustitution) Firstly, we n turn the proof of (.) into the inequlity (.) s ove. Sine. r, (.5) Mthetis Setion

5 On n inequlity for Jin Liu thus inequlity (.) is equivlent to i.e.,, (.6) Y 4. (.7) Let x, y, z, then y z, z x, x y, nd we hve Y y z y z x y y z yz zx zx xy x y z z x x y 4. (.8) Beuse of syetry, we ssue without loss of generlity tht x y z nd let y z x z n. (.9) where nd n. Sustituting (.9) into (.8), with help of the thetil softwre we otin the following identity: 8 7 Y 4 ( nn ) z 56 ( n)( nn )( n6 n ) z ( n 7968 n 658 n 96 n 864 n n ) z 8 ( n )(8 54 n 45 n n n n n ) z 448 n 496 n 859 n n 4 n 6976 n 768 n n ) z (.) ( n)( n )( n 5 n 8 n 79 n n n ) z 6 ( n 67 n 94 n 6 n n n )( n) z 6 ( n n )( n) ( n) z ( n) ( n). So inequlity Y holds oviously y, n, nd z >. Hene (.7), (.6) nd then (.) re proved. The equlity in Y holds if nd only if = n =. Further, it is nown tht the equlities in (.7) nd (.) ours only when = =, i.e. ΔABC is equilterl. This opletes the proof of Theore. er.. Fro inequlity (.), using previous ethods to prove Theore we n lso prove the following inequlity: ISSN: Mthetis Setion

6 On n inequlity for Jin Liu whih is posed y the uthor in []. 4 r 4 4, (.). TWO APPLICATIONS OF THEOEM In this setion, we will pply Theore to estlish two new tringle inequlities, whih re not oth proved y using inequlity (.). We first prove the following eutiful liner inequlity: Theore A. For ll ΔABC holds: h h h r, (.) with equlity if nd only if ΔABC is equilterl. Proof: By Theore, to prove (.) we need to prove tht r h h h r. (.) Multiplying oth sides of this inequlity y 4 nd using the reltion h = et., inequlity (.) eoes the following equivlent for: M 4 r 4 r. (.) Applying identity (.6) nd the nown identity: it is esy to get s 4r r, (.4) M r r s. (4 4 ) Thus the lied inequlity M follows nd (.) is proved. It is ler tht the equlity in (.) holds only when ΔABC is equilterl. This opletes the proof of Theore A. er.. By the ethod to prove Theore in [], we n prove tht the onstnt of the right side of (.) is the est possile. In ddition, fro Leuenerger s inequlity (see [6]): h h h r, (.5) 5 we see tht inequlity (.) is stronger thn the nown result (see [6]): Mthetis Setion

7 On n inequlity for Jin Liu 4 r. (.6) er.. By using inequlity (.), it is esy to prove nother liner inequlity for the su : s 6 9r. (.7) This inequlity is lso stronger thn (.6) sine we hve the following inequlity: s 4r, (.8) whih is due to W.J.Blundon (see [7], [8], [9]). Next, we prove n inequlity for the ute-ngled tringle, whih ws found y the uthor ny yers go, ut hs not een proved efore. Theore B. For ute-ngled ΔABC holds: h h h r, (.9) with equlity if nd only if ute-ngled ΔABC is equilterl. Proof. By Theore, to prove (.9) we need to show tht r r. (.) h h h Multiplying oth sides of the ove y we see (.) is equivlent to 4 4 nd then using the reltion N r r h et.,. (.) Sustituting (.6) nd (.4) into the expression of N, then (.) is equivlent to N s r r r r s r 4 r9 r 8r 8r r. (.) We rewrite N s follows 4 N 4r r r e8r G 4r r eg C, (.) where e r G s 6r r 5 ISSN: Mthetis Setion

8 4 On n inequlity for Jin Liu G 4 4r r s C s r. Therefore, y Euler inequlity e, Gerretsen inequlities G, G (see [5], [6]) nd the ute tringle inequlity C of Cierlini (see []), we onlude N holds for ute-ngled ΔABC. Hene inequlity (.) nd (.9) re proved. It is esy to see tht the equlity in (.9) holds when ΔABC is equilterl. The proof of Theore B is opleted. 4. SEVEAL CONJECTUES In this setion, we will propose soe onjetures for the inequlities ppered in this note. Considering the exponentil generliztion of Theore A with help of the oputer for verifying, we pose the following three siilr onjetures: Conjeture. If, then for ny ΔABC we hve h h h 4r (4.) If ΔABC is n ute tringle nd., then the inequlity holds reversed. er 4.. It is esy to prove tht (4.) is reversed for ll tringles if <. Conjeture. If ΔABC is n ute tringle nd., then we hve h h h r. (4.) Conjeture. If > or <, then for ny ΔABC we hve 4 h h h r When =, (4.) is tully equivlent to. (4.), (4.4) 4r whih is ler weer thn the nown inequlity (see []): w, (4.5) w w 4r where w, w, w re three internl isetors of ΔABC. On the other hnd, (4.4) n e refined the following: Mthetis Setion

9 On n inequlity for Jin Liu 5, r (4.6) whih is proved y the uthor in []. Considering the lower ound of the left hnd side of (.), we give Conjeture 4. For ny ΔABC we hve h h h s r. (4.7) If (4.7) holds true, then Blundon s inequlity (.8) n e otin fro (.) nd (4.7). Next, we give doule inequlity onjeture whih is inspired y Theore B: Conjeture 5. For ny ΔABC we hve r r r r, (4.8) where,, re syedins of ΔABC nd r, r, r re rdii of exirles of ΔABC. Considering the exponentil generliztion of inequlity (.), we present Conjeture 6. If >, then for ny ΔABC we hve r. (4.9) If 8, then the inequlity is reversed. 5 The lssil Erdös-Mordell inequlity n e stted s follows: Let P e n interior point of ΔABC. Denote y,, the distnes of P fro the verties A, B, C, nd r, r, r the distnes of P fro the sidelines BC,CA,AB respetively. Then holds: r r r. (4.) It is well nown tht there re few stronger versions of the Erdös-Mordell inequlity (see e.g. [5], []). Here, we put forwrd two new stronger inequlities. Conjeture 7. For ny interior point of ΔABC, we hve r r r 4s 6r 5 r (4.) Inequlity (.) shows (4.) is stronger thn (4.). Conjeture 8. For ny interior point of ΔABC, we hve ISSN: Mthetis Setion

10 6 On n inequlity for Jin Liu r r r h h h. 4r (4.) The following equivlent for of (.): h h h r (4.) 4 ens gin (4.) is stronger thn the Erdös-Mordell inequlity (4.). EFEENCES [] Chu, X.-G., Yng X.-Z., Soe inequlities for the edins of tringle, eserh in Inequlities, Tiet People s Press,. [] Liu, J., Chun X.-G., Journl of Chin Jiotong University, (), 89,. [] Liu, J., Trnsylvnin Journl of Mthetis nd Mehnis, (), 4,. [4] Yin, H.-Y., onjeture inequlities involving Cev segents nd rdii of tringle, eserh in Ineqlities, Tiet People s Press, Lhs,. [5] Mitrinović, D.S., Pečrić, J.E., Volene, V., eent Advnes in Geoetri Inequlities, Kluwer Adei Pulishers, Dordreht-Boston-London, 989. [6] Botte, O., Djordjević,.Z., Jnić,.., Mitrinović, D.S., Vsić, P.M., Geoetri Inequlities, Groningen, 969. [7] Blundon, W.J., Cn. Mth. Bull, 8, 65, 965. [8] Dospinesu, G., Lsu, M., Pohot C., Tetiv, M., J. Inequl. Pure Appl. Mth., 9(4), Art., pges, 8. [9] Stnoinu,.A., Mth. Inequl. Appl., 7(), 89, 4. [] Cierlini,C., Bull.Un.Mt.Itl., 5(), 7, 94. [] Chu, X.-G., Liu, J., Missouri Journl of Mthetil Sienes, (), 55, 9. [] Liu, J., Journl of Est Chin Jiotong Univeresity, 5(), 5, 8. [] Liu, J., Int. Eletron. J. Geo, 4(), 4,. Mthetis Setion