Some Inequalities Between Two Polygons Inscribed One in the Other

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1 BULLETIN of the Malaysan Mathematcal Scences Socety Bull. Malays. Math. Sc. Soc. () 8() (005), Some Inequaltes Between Two Polygons Inscrbed One n the Other Aurelo de Gennaro Vale Molse,, Larno (CB), Italy adegennaro@fdespa.com Abstract. It s well known that, gven a trangle nscrbed n another trangle, the permeters of the three external trangles can never all be smultaneously greater than the permeter of the nscrbed trangle and that furthermore they are all equal to t f and only f we put the vertces of the nscrbed trangle at the mdponts of sdes of the crcumscrbed trangle. The same result s true for the areas. The present paper shows how such a results extends to the case of two convex polygons nscrbed one n other, connectng t to the classc works about nscrbed and crcumscrbed polygons respectvely wth mnmum and maxmum permeter. 000 Mathematcs Subject Classfcaton: 5M6 Key words and phrases: Geometrc nequaltes, extremum problems.. Introducton The followng neat nequalty between two trangles one nscrbed n the other was proposed by Debrunner [] n 956 and proved frst by Dresel [] n 96 and later on, wth dfferent methods, by other mathematcans: Theorem.. Gven a trangle ABC and any other trangle A B C crcumscrbng t, t s not possble that we have smultaneously: P(ABC) < P(A CB), P(ABC) < P(CB A) and P(ABC) < P(AC B) where P(ABC) ndcates the permeter of trangle ABC. Another result of the same knd, also proposed by Debrunner [3] n 956 and proved by Bager [4] n 957 and later on by other mathematcans, s the followng: Theorem.. Gven a trangle ABC and any other trangle A B C crcumscrbng t, t s not possble that we have smultaneously: S(ABC) < S(A CB), S(ABC) < S(CB A) and S(ABC) < S(AC B) where S(ABC) ndcates the area of trangle ABC. Receved: December 7, 000; Revsed: December 8, 003.

2 74 A. de Gennaro In ths paper we want to show how the nequaltes can be extended to the case of two convex polygons nscrbed one n the other, thus achevng a result that generalzes Theorems. and... The general case of convex polygons In the case of convex polygons the followng s n fact true: Theorem.. Gven a convex polygon of n sdes wth vertces A (,...,n) numbered cyclcally so that A 0 A n, and any other polygon of n sdes crcumscrbng t wth vertces A (wth A lyng on the sde A A ), f we call B the ntersecton of the dagonals A A + and A A + (see Fg. ), t s not possble that we have smultaneously: (.) P(A B A + ) < P(A A + A ) for,...,n or (.) S(A B A + ) < S(A A + A ) for,..., n. Furthermore one and only one crcumscrbed n gon exsts such that P(A B A + ) P(A A + A ) and S(A B A + ) S(A A + A ) for,...,n, and that s acheved by puttng the sde A A + parallel to the dagonal A A +. We defne the angular coordnate α (for,...,n) as the angle whch the sde A A makes wth the lne parallel to dagonal A A + passng for vertex A (see Fg. ) and fxng that the parallel lne has to be rotated counterclockwse through a postve angle α n order to become concdent wth A A. For the law of snes we wll have that: (.3) A B + B A + sn λ + snµ + A A + sn(λ + µ + ) for,..., n wth λ t s evdent that so that: (.4) λ+µ+ sn cos λ µ+ sn λ+µ+ cos λ+µ+ cos λ µ+ cos λ+µ+ A + A A + and µ A A A (for,...,n). Now A A + A λ + α + and A A + A A + sn(µ + α ) + sn(λ + α + ) A A + sn(λ + µ + + α + α ) A+ A A µ + α (see Fg. ) and for,...,n. Now, wth α as defned above, t s evdent that: (.5) λ < α < µ + for,...,n cos λ µ++α ++α cos λ+µ++α+ α and the denomnators of last term of (.3) and (.4) are both postve. Hence P(A B A + ) P(A A + A ) wll have the same sgn as: (.6) cos λ µ + cos λ + µ + + α + α cos λ + µ + cos λ µ + + α + + α ( sn λ + α ) + sn α ( sn µ + α ) sn α +.

3 Some Inequaltes Between Two Polygons Inscrbed One n the Other 75 A D D + A + α + D + A + µ + λ A A + µ + B α j A λ D A Fgure. General case of convex polygons There are now three possbltes: Case (): The α are all zero. Then, for the (.6), 0 for,...,n and the sde A A + s always parallel to the dagonal A A +. Case (): There s only one α 0 or there are two that don t have the same sgn. In such cases, on the varyng of, there wll be an for whch α > 0 and α + 0 (or α 0 and α + < 0). But for (.5) µ + α > 0 and λ + α+ > 0 and so, on the bass of (.6), 0 and the theorem s proved. Case (3): All the α, for,...,n, are dfferent from zero and have the same sgn. In such a case, dvdng (.6) by sn α α+ sn (> 0), has the same sgn as: (.7) sn ( ) λ + α+ sn α+ sn λ cotg α + sn( µ + α sn α ) sn µ + cotg α + cosλ + cosµ +

4 76 A. de Gennaro and supposng ab absurdo that < 0 for,...,n and consderng that cosλ + cosµ + > 0 we have: (.8) snλ cotg α + < snµ + cotg α for,...,n. Now f α > 0 for,...,n (smlar argument also holds f α < 0) and dvdng by snλ cotg α (> 0) we have: α+ cotg (.9) 0 < < sn µ + for,..., n. cotg α sn λ Multplyng (.9) for,..., n and rememberng that we wll have: (.0) < sn µ + sn λ sn µ + sn λ snµ +, sn λ sn λ +µ+ cos λ µ+ cos λ +µ+ sn λ µ+ sn λ +µ+ cos λ µ+ + cos λ +µ+ sn λ µ+ and dvdng numerator and denomnator by sn λ +µ+ cos λ µ+ (> 0) and utlsng the theorem of tangents n the trangle A A A + we have: (.) < cotg λ +µ+ tg λ µ+ + cotg λ +µ+ tg λ µ+ AA+ A A A A ++A A + AA+ A A A A ++A A whch s a contradcton. To prove (.) we observe, frst of all, that: (.) S(A B A + ) A A + sn µ + sn λ sn(µ + + λ ) and: A A A A + (.3) S(A A + A ) A A + sn(µ + α )sn(λ + α + ) sn(µ + + λ + α + α ) for,...,n for,...,n and, snce sn(µ + + λ )sn(µ + + λ + α + α ) > 0, then S(A B A + )+ S(A A + A ) has the same sgn as: (.4) sn µ + sn λ sn(µ + + λ + α + α ) sn(µ + α )sn(λ + α + )sn(µ + + λ ) sn µ + sn λ [sn(µ + α )cos(λ + α + ) + cos(µ + α )sn(λ + α + )] sn(µ + α )sn(λ + α + ) [sn µ + cosλ + cosµ + sn λ ].

5 Some Inequaltes Between Two Polygons Inscrbed One n the Other 77 Dvdng by snµ + sn λ sn(µ + α )sn(λ + α + )(> 0), (.4) becomes: (.5) cotg (λ + α + ) cotg λ + cotg (µ + α ) cotg µ +. Let s examne the three cases analysed before: Case (): If α are all zero then, from (.5), 0 for,...,n. Case (): If there s an such that α > 0 and α + 0 (or α 0 and α + < 0) then, for (.5), > 0 and the result s proved. Case (3): If fnally all the α are dfferent from zero and wth the same sgn, developpng (.4), we acheve that has the same sgn as: (.6) sn µ + sn λ snα [sn λ cosα + + cosλ sn α + ] + cos µ + snλ sn α [sn λ cosα + + cosλ snα + ] sn µ + sn λ sn α + [snµ + cosα cosµ + sn α ] sn µ + cos λ sn α + [sn µ + cosα cosµ + sn α ] that, dvdng by snα sn α + (> 0), become: (.7) cotg α + sn λ cotg α sn µ + + sn µ + cosµ + + sn λ cosλ. Now, supposng ab absurdo that < 0 for,...,n and consderng that snµ + cosµ + + sn λ cosλ > 0, we wll have that: (.8) cotg α + sn λ < cotg α sn µ + for,...,n and, supposng α > 0 for,...,n (f α < 0 smlar argument holds but reversng the nequalty), we have: (.9) 0 < cotg α + cotg α < sn µ + sn λ for,...,n and, multplyng (.9) for,...,n, we wll have, for the sne rule, that: sn µ + A A (.0) < sn λ A A + that s agan a contradcton. 3. Conclusve consderatons It s clear that f n 3 the mnmum dagonals A A + concde wth the sdes of trangle A A A 3, the ponts B wth the opposte vertces and Theorem. s reduced to Theorems. and.; Theorem. beng thus a generalzaton of Theorems. and.. Another way to express Theorem. s the followng. Takng the convex polygon wth vertces A (,...,n) we consder the ellpses whch have foc n two subsequent vertces A and A + and whch pass through the pont of ntersecton B. If we now magne throwng a ball from one of the vertces, towards the outsde and beyond the external angle opposte the vertex of the angle of the polygon, and we bounce t subsequently on the n ellpses we wll see that t wll return to the ntal vertex and that the drecton of departure and that of arrval, seen from the nsde of polygon, wll always form an angle π.

6 78 A. de Gennaro A +3 A + B + B + A + B B A B B A B B A A 3 Fgure. Internal and external polygons It s nterestng to remark too that the ponts B also form a convex n gon whch corresponds, from a projectve pont of vew, to the dual n gon B made by ntersectons of sdes A A and A + A + of polygon A adjacent to the same sde (see Fg. ) and that to Theorem.: P(A B A + ) P(A A + A ) corresponds to the dual result P(A B A +) > P(A A + A ) and lkewse for areas. In substance the result acheved tell us that the n vertces of crcumscrbed polygon A cannot smultaneously (n a manner of speakng) dstance themselves from the correspondng sdes of the nscrbed polygon A A + further than the correspondng ponts B are dstant from that sde. The result s n fact, n a manner of speakng, connected to the classc works about mnmum and maxmum permeters of respectvely nscrbed and crcumscrbed polygons represented by the well known (see e.g. [5]). Theorem 3.. The n gon wth vertces A nscrbed n a gven n gon wth vertces A (,...,n) wth mnmum permeter n all t s that n whch µ + α λ + α for,..., n. Usng the notatons ntroduced before we can say that the arthmetc mean of s maxmum for α µ+ λ for,...,n and that nstead, n order to maxmze

7 Some Inequaltes Between Two Polygons Inscrbed One n the Other 79 the mnmum of and of, the followng wll be true α 0 for,...,n and n such a case wll be mn{,...,n} mn{,...,n} 0. Of course fnally f the n gon A s regular the two nscrbed n gons wll concde. References [] O. Bottema, R. Djordjevć, R. R. Janć, D. S. Mtrnovć and P. M. Vasć, Geometrc Inequaltes, Wolters-Noordhoff, Gronngen (969), 83. [] L. A. G. Dresel, Nabla, Bull. Malays. Math. Soc. 8 (96), [3] H. Debrunner, Elem. Math. (956), 0, Problem 60. [4] A. Bager, Elem. Math. (957), 43. [5] N. D. Kazarnoff, Geometrc Inequaltes, Yale Unversty, (96).