On Trigonometrical Proofs of the Steiner-Lehmus Theorem

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1 On Trigonometricl Proofs of the Steiner-Lehmus Theorem Róbert Oláh-Gál nd József Sándor Dedicted to the memory of Professor Ferenc Rdó ) Abstrct We offer survey of some less known or new trigonometricl proofs of the Steiner-Lehmus theorem. A new proof of recent refined vrint is pointed out, too. 1 Introduction The fmous Steiner-Lehmus theorem sttes tht if the internl ngle bisectors of two ngles of tringle re equl, then the tringle is isosceles. For recent survey of the Steiner-Lehmus theorem, see M. Hjj [6]. From the References in [6] one cn find mny methods of proof of this theorem, including pure geometricl, trigonometricl, etc. proofs. One im of this note is lso to dd some new references; to cll the ttention to some little or not-known proofs, especilly trigonometricl ones. On the other hnd, we will obtin lso new trigonometric proof of refined version of the Steiner-Lehmus theorem, published recently [7]. For the first im, we wnt to point out some clssicl geometricl proofs published in 1967 by A. Frod [], ttributed to W.T Willims nd G.T. Svge. Another interesting proof by A. Frod ppers in his book [5] see lso the book of the second uthor [1]). Another pure-geometricl proof ws published in 1973 by M. K. Sthy Nrym [13]. Other ppers re by K. Seydel nd C. Newmn [1], or the more recents by D. Bern [1] or D. Rüthing [10]. None of the recent extensive surveys connected with the Steiner-Lehmus theorem mentions the use of complex numbers in the proof. Such method ppers in the pper by C.I. Lubin [8] from Relted to the question, first posed by Sylvester nd mentioned in [6], too) whether there is direct proof of the Steiner-Lehmus theorem, recently J. Conwy see []) hs given n intriguing rgument tht there is no such proof. However, there re discussions on the vlubility of this proof, s perhps we should formulte in completely precise mnner this proposition: the Steiner-Lehmus theorem hs no direct proof by using e.g., concept of intuitionistic logic) 1

2 Trigonometric proofs of the Steiner-Lehmus theorem Perhps one of the shortest trigonometric proofs of the Steiner-Lehmus theorem one cn find in forgotten pper written in Romnin) from 1916 by V. Cristescu [3]. Let BB nd CC denote two ngle bisectors of the tringle ABC see fig. 1). Figure 1: By using the sinus theorem in tringle BB C, one gets BB sin C = As C + B = C C A = 90 A C, one hs BB = One cn obtin in similr mnner the reltion CC = BC sin C + B ). sin C A C. 1) sin B A B. ) Assuming BB = CC, nd remrking tht sin C = sin C C, nd sin C = A + B, sin B = A + C, we get the equlity C A + B Now from the identity A B = B A + C A C. 3) x + y) x y) = x + y 1, )

3 reltion ) becomes C A + B ) 1 = B A + C 1 ). 5) A simple remrk shows tht 5) cn be rewritten lso s B C ) sin A + B C ) = 0. 6) As the second prnthesis of 6) is strictly positive, this implies B C = 0, so B = C. In 000, resp. 001 the Germn mthemticins D. Plchky [9], nd D. Rüthing [11] hve given other direct trigonometric proofs of the Steiner-Lehmus theorem, bsed on re considertions. We will present here shortly the method by D. Plchky [9]. Denote the ngles from B nd C resp. by β nd γ, nd the ngle bisectors BB nd AA by w b nd w see fig.). Figure : By using the trigonometric form of the re of tringle ABC s 1 b sin γ, nd decomposing the initil tringle in two tringles, we get 1 w β sin β + 1 cw β sin β = 1 bw α sin α + 1 cw α sin α. By the sinus-lw we hve sin α = sin β b so ssuming w α = w β, we obtin = sinπ α + β)), c sin α c sinα + β) sin β + c sin β = c sin β sinα + β) sin α + c sin α, 3

4 or sinα + β) sin α sin β ) + sin α sin β sin α sin β = 0. 7) Writing sin α = sin α α, etc; nd using lso the formule sin u sin v = sin u v we get from 7) sin α β [sinα + β) α + β u + v, u v = sin u v sin u + v, 8) + sin α sin β sin α + β ] = 0. 9) As in 9) the prnthesis is strictly positive by 0 < α + β < π, 0 < α+β < π), 9) implies α = β. The following trigonometric proof due to the uthors) seems to be much simpler. Writing the re of tringle ABC in two distinct wys using tringles ABB nd BB C) we get immeditely Similrly, w b = w = c + c β. 10) bc b + c α. 11) Suppose now tht, > b. Then α > β, so α > β. As α, β 0, π ), one gets α < β bc. Remrk lso tht b + c < c is equivlent to b <. + c Thus 10) nd 11) imply w > w b. This is indeed proof of the Steiner-Lehmus theorem, s supposing w = w b nd letting > b, we would obtin w > w b, contrdiction; nd if < b, then w < w b, gin contrdiction. 3 A new trigonometric proof of refined version Recently, M. Hjj [7] proved the following stronger version of the Steiner- Lehmus theorem. Let BY nd CZ be the ngle bisectors nd denote BY = y, CZ = z, Y C = v, BZ = V see fig. 3). Then c > b y + v > z + V. 1) As V = c + b, v = b, it is immedite tht c > b V > v. + c Thus, ssuming c > b, on bse of 1) we get y > z, i.e. the Steiner-Lehmus theorem see the lst proof of prgrph ). In the proof of 1), in [7] strong lemm by R. Breuch is pplied.

5 Figure 3: Our im here is to offer new trigonometric proof of 1), bsed only on the sinus-lw, nd simple trigonometricl fcts. In tringle BCY one cn write sin C + B ) = CY sin B = BY sin C, so y + v = y + v sin C + sin B = sin C + B ) implying, sin C + sin B ) sin C + B ). 13) In completely similr mnner one gets sin B + sin C ) z + V = sin B + C ). 1) Assume now tht y + v > z + V. Applying sin u + sin v = sin u + v u v C nd remrking tht + B ) B > 0, + C ) > 0, fter simplifiction, from 13) 1) we get the inequlity C B ) B + C ) B > C ) C + B ). 15) Using u v = u + v + u v, this implies 3C + B ) C + 3B ) 3B > + C ) B + 3C 5 ),

6 or 3C + B ) 3B + C ) B > 3C Now pplying the second formul of 8), we get ) C 3B ). 16) sin B sin 3C > sin C sin 3B. 17) By sin 3u = 3 sin u sin 3 u we get immeditely from 17) tht 3 + sin C > 3 + sin B. 18) Remrk now tht the function x sin x is strictly incresing in x so s 18) gives sin C > B sin, this is possible only if 0, π ), C > B. 19) This finishes the proof of 1), s if the impliction in 1) would not be true, then the rgument bove would imply C B, contrry to c > b. Acknowledgements. The uthors thnk Professor D. Plchky for reprint of [9] nd to Professor A. Furdek for providing copy of [11]. References [1] D. Bern, SSA nd the Steiner-Lehmus theorem, Mth. Techer, 85199), no. 5, [] J. Conwy, PUZZLES/steiner_lehmus [3] V. Cristescu, On problem of geometry Romnin), Gzet Mtemtică Bucureşti), 1916),no., [] A. Frod, Nontrivil spects of certin questions of eucliden geometry Romnin), Gzet Mtemtică, 71967), no. 1, 1 7 [5] A. Frod, Errors nd prdoxes in Mthemtics Romnin), Ed. Enc. Rom., Bucureşti, 1971 [6] M. Hjj, A short trigonometric proof of the Steiner-Lehmus theorem, Forum Geometricorum, 8008), 3 [7] M. Hjj, Stronger forms of the Steiner-Lehmus theorem, Forum Geometricorum, 8008),

7 [8] C. I. Lubin, The theorem of Lehmus nd complex numbers, Script Mth., 1959), [9] D. Plchky, Ein direkter und elementr-trigonometrischer Beweis des Stzes von Steiner-Lehmus und trigonometrisch-geometrische Lösung einer Vrinte. Preprints Universität Münster,,Angewndte Mthemtik und Informtik, /000, S.5 [10] D. Rüthing, Ein einfcher elementr geometrischer Beweis des Stzes von Steiner-Lehmus, Prxis Mth. Heft, /000, S.176 [11] D. Rüthing, Ein direkter trigonometrischer Beweis des Stzes von Steiner- Lehmus, Wurzel, 7/001, [1] J. Sándor, Geometric inequlities Hungrin), Ed. Dci, Cluj, 1988 [13] M. K. Sthy Nrym, A proof of the Steiner-Lehmus theorem, Mth. Student, XLI, no. 1973), [1] K. Seydel nd C. Newmn, The Steiner-Lehmus theorem s chllenge problem, The Two-Yer College Mth. J., ), no. 1, 7 75 Róbert Oláh-Gál nd József Sándor: Deprtment of Mthemtics nd Informtics, Bbeş Bolyi University, Extension of Miercure Ciuc, Romni Str. Topliţ Nr. 0. Miercure Ciuc. E-mil ddresses: olh gl@topnet.ro jsndor@mth.ubbcluj.ro 7