BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. () 8() (005), 73 79 Some Inequaltes Between Two Polygons Inscrbed One n the Other Aurelo de Gennaro Vale Molse,, 86035 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.
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 +.
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 +
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 ].
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.
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
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), 97 99. [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).