Inequalities for the internal angle-bisectors of a triangle

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1 Mtheticl Counictions 2(997), Inequlities for the internl ngle-bisectors of tringle Wlther Jnous nd Šefket Arslngić Abstrct. Severl ne inequlities of type α ± for ngle-bisectors re proved. Certin lgebric cyclic inequlities re derived. To conjectures nd n open question re entioned. Key ords: geoetric inequlities, ngle-bisectors, cyclic inequlities. Sžetk. Dokzno je nekoliko novih nejednkosti tip α ± z sietrle kutov. Izvedene su neke cikličke nejednkosti. Spoenute su dvije pretpostvke i jedno otvoreno pitnje. Ključne riječi: geoetrijske nejednkosti, sietrle kutov, cikličke nejednkosti AMS subject clssifictions: 5M6 Received Mrch, 997, Accepted June 6, 997. Introduction In this note e extend ethods fro 2] in order to get soe further inequlities linking the ngle-bisectors nd other eleents of (plnr) tringle. We lso stte soe conjectures. As usul, for tringles, b, c denote the internl ngle-bisectors nd, b, c, s, r nd R the sides, the seiperieter, the inrdius nd the circurdius, resp.. All sus ppering re cyclic. (e.g. bc ens b + bc + c, etc.) Throughout this note e let > 0 be rel nuber. 2. Soe let In this section e shll estblish tringle-inequlities of type k (bc), here k := k (s, R) is best possible. Due to the fct tht for, sy, c fixed nd Ursulinengynsiu, Fürsteneg 86, A-6020 Innsbruck, Austri Prirodno-tetički fkultet, Zj od Bosne 4, BiH-7000 Srjevo, Bosni nd Hercegovin

2 42 W. Jnous nd Š. Arslngić = b c/2 e get R, there re fctors k fvourble not depending on R.] We firstly prove the lgebric Le. Let x, y nd z be nonnegtive rel nubers such tht x + y + z 0. Then Proof. We hve xy + yz + zx x + y + z x + y + z () (xy + yz + zx) (x + y + z) 2 xy + yz + zx x 2 + y 2 + z 2 (x y) 2 + (y z) 2 + (z x) 2 0 ) k is of type K /s, K best possible constnt. Then there holds the folloing Le 2. K = (/2) for 0 <. I.e., if 0 < then /(2s)] (bc). Proof. Strting fro (/) /(2s)] = e get vi the generl ensinequlity (/) /(2s)]. Putting in Le x = /(2s)] etc. e get b/(2s)] c/(2s)] /(2s)] (/) /(2s)] nd thus /(2s)] (bc), s clied. Rerk. It reins n open question to deterine the vlue of K in cse >. Coputer-serches indicte the folloing Conjecture.. Let µ := ln 2/(ln ln 2) = Then { (/2) K =, if 0 < µ 2, if > µ () b) k is of type L /R, L best possible constnt. Then e hve the folloing Le. L = (/ ) for 0 < τ, here τ := (ln 9 ln 4)/(ln 4 ln ) = I.e., if 0 < τ then /(R )] (bc) Proof. ] hs s ite 5.28 tht the inequlity (/) /(R )] holds true if nd only if 0 < τ. The rest of the proof of Le is siilr s for Le 2. Rerk 2. Agin it reins n open question to deterine L in cse > τ. For this e stte the folloing Conjecture 2.. Let σ := ln 4/(ln 4 ln ) = Then { (/ ) L =, if 0 < σ /2, if > σ It should be noted tht due to ], ite 5., i.e. 2s R, for Le 2 lys yields the better fctor k thn Le (in cse of vlidity of Conjecture even for µ). On the other hnd, if both conjectures re vlid, for > µ it depends on the shpe of tringle hether Le 2 or yields the better fctor k.

3 Internl ngle-bisectors inequlities 4 c) An ll-over loer estition of the fctor k = k (s, R). We shll no prove Le 4. k > x {/s, /(2R) } for ll > 0. I.e., if > 0 then > x {/s, /(2R) } (bc) Proof. Due to, b, c 2R nd, b, c < s e get, b, c in(s, 2R) =: M ith t lest one side < M. Hence (bc) < M nd the clied inequlity follos. Rerk. In vie of ) nd b) Le 4 deserves ttention if >.. The results A) Cse. We re no in the position to prove the folloing Theore. Let. Then s 2 + (4R + r) 2 ] 2Rrs (2) 2bc cos(a/2) Proof. We strt fro the ell-knon forule = b+c, etc. Since inequlity (2) is syetric ith respect to, b nd c e y nd do let b c. Then i) /c /b / nd further /b + /c /c + / / + /b nd ii) A B C nd thus / cos(a/2) / cos(b/2) / cos(c/2). Using Chebyshev s nd the rithetic-geoetric inequlities e get = cos (A/2) ( 2 b + )] 2 c cos (A/2) ( 2 b + )] 2 c cos (A/2) bc The generl ens-inequlity nd ], p. 5, yield ] s cos (A/2) cos (4R + r) 2 (A/2) = s 2 This inequlity nd the observtion ] = bc (bc) = (4Rrs) ] ] () ieditely vi () led to the clied inequlity (2). Applying Le e get fro (2) the folloing Corollry. Let τ. Then s 2 + (4R + r) 2 ] s 2 R. (4) Using ], ites 5.5 nd 5.2, i.e. (4R + r) 2 s 2 nd Rr 2s 2 /27, resp., e get fro (4) nd (2) the further

4 44 W. Jnous nd Š. Arslngić Corollry. here τ nd Corollry 2. ] 4 R, (5) ] ] 9 Rrs 2s, (6) here. Rerk 4. For = e get fro (2) nd (6) the folloing iproveent (interpoltion) of 4], p. 27, ite.: 2 s2 + (4R + r) 2 6Rrs 2 2 Rr 9 s 2. B) Cse 0 < < (0 < ). In this cse there cn hold only (clerly) eker inequlity stted s Theore 2. 4Rr s 5 ] / Proof. Proceeding s for Theore e get fro () nd ], p. 5, vi the rithetic-geoetric inequlity Π cos(a/2) ] 2 ] = bc ( ) 2 4R s (bc), nd the clied inequlity follos. We no dd soe consequences of Theore 2. Vi Le 2 e get Corollry. ( ) 54R 2 (7) Using once ore R 2s inequlity (7) leds us to Corollry 4. ( ) 2 s s 5 Rerk 5. The bove proofs ieditely iply tht equlity holds in either of the inequlities if nd only if = b = c.

5 Internl ngle-bisectors inequlities Algebric nlog of inequlities (6) nd (7) In this section e ill pply the useful trnsfortion = x 2 + x, b = x + x nd c = x + x 2, here x, x 2 nd x re rbitrry positive rel nubers (see e.g. ], chpt. II). It lys links geoetric nd lgebric (three-vrible) inequlities. Then upon norlizing (s =) x + x 2 + x = short siplifictions yield = ( + x )/(2 x ( x 2 )( x )), etc. nd R = bc/4f = ( x )( x 2 )( x )/(4 x x 2 x ). Hence e get the folloing lgebric nlog of Corollries nd k= k= ( + ) 2 x ] ( ) k 4 ( x x 2 x ) here τ, nd ( ) ( + ) 2 ] { ] } 6 xk here 0 < < (0 < ). k= Both inequlities led to the folloing k= ( ), ( x i ) ( x j ) Open Questions. Deterine the respective sets A nd B of ll nturl nubers n such tht the n-diensionl nlog re vlid ith throughout ll x,..., x n (0, ), x +...+x n =, nd siilrly, resp., nd the respective constnt fctors re replced by the suitble constnts yielding equlity t x =... = x n (= /n). References ] O. Botte, R. Z. Djordjević, R. R. Jnić, D. S. Mitrinović, P. M. Vsić, Geoetric Inequlities, Wolters-Noordhoff, Groningen ] W. Jnous, An inequlity for the internl ngle-bisectors of tringle, Univ. Beogrd Publ. El. Fk. Ser. Mt. 7(996), ] W. P. Soltn, S. I. Mejdn, Identities nd Inequlities for Tringles, in Russin], Stiinc, Kishinev, ] D. S. Mitrinović, J. E. Pečrić, V. Volenec, Recent Advnces in Geoetric Inequlities, Kluer, Dordrecht, 989. i<j