On Some Maximum Area Problems I

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2 On Som Maximum Ara Problms I 1. Introdution Whn th lngths of th thr sids of a triangl ar givn as I 1, I and I 3, thn its ara A is uniquly dtrmind, and A=s(s-I 1 )(s-i )(s-i 3 ), whr sis th smi-primtr t{i 1 + I + I 3 ). This formula is usually alld th Hron's formula. It is also wll known that thr is a uniqu irl irumsribing any givn triangl. Howvr, whn th numbr of sids of a plan polygon is mor than thr, th situation boms muh mor ompliatd. For instan, onsidr all quadrilatrals of whih th lngths of th four sids ar givn as I 1, I, I 3, and I 4 Sin thr ar infinitly many suh quadrilatrals, rathr than onsidring th ara of an individual quadrilatral, it is mor natural to addrss th following qustions: (1) Whih of ths quadrilatrals is yli? What is th radius of th irumsribing irl? () Whih of ths quadrilatrals ahivs th maximum ara? What is th formula of th maximum ara? Is suh a quadrilatral uniqu? In gnral, for any positiv intgr n 5, similar qustions an also b posd for n-gons with n sid lngths prsribd. In this papr, w shall prsnt to radrs th main rsults prtaining to ths qustions in a mor systmati way, and try to provid th solutions to th problms in an asily assibl way. W shall first answr th qustions mntiond abov for quadrilatrals, thn prov som rsults for gnral n-gons. To undrstand th disussion, th radrs only nd to hav basi knowldg of alulus..... Conditions For A Quadrilatral To Ahiv Maximum Ara Sin w ar mainly intrstd in thos polygons whih ahiv maximum ara and suh polygons ar larly onvx, thus in th following w shall only onsidr onvx polygons. W shall first onsidr th quadrilatral whih ahivs th maximum ara. Lt ABCD b any quadrilatral whos four sids hav lngths Ipi,I 3 and I 4 (s Figur 1). Join AC and lt and If! dnot th angls L ABC and L ADC, rsptivly.

3 Volum 9 No., Dmbr Thus, th ara A of th quadrilatral ABCD is givn by: (1) In viw of AC =1 1 +1/ os = os If/, or os = k 1 + k oslfl, (). A 14 lz whr D If/ B k = ' I. ' 1>, ;u E Cl) s::: fl} --.!:J «S = ::;: '' 13 1 Figur 1 Substituting (3) into (1), w obtain Now A is a ontinuous funtion of th variabl If! and it is diffrntiabl ovr (, tr). In ordr to find out th valu of 1f1 for whih A attains th maximum valu, w first find its stationary point(s) in (, tr). Now Using (3) and (4), 1 3 *, 1 4 *, and da = w obtain dljl In viw of (), (5) boms sin os If!+ os sin If! =, that is, sin( + If!)=. (3) (4) (5) (6) t',

4 Sin < < 1& and < lfl < 1&, quation (6) implis +lfl=1&. (7) From () it follows asily that thr is a uniqu lfl satisfying (7), that is A has a uniqu stationary point in (, n). Furthrmor, for this lfl* A (8) D B Thus from lmntary alulus, it follows that A attains th absolut maximum valu at lfl. Figur Noti that quation (7) is atually quivalnt to th ondition that th quadrilatral ABCD is yli, i.., all its vrtis li on a irl as shown in Figur. In th following, w shall show that whn and lfl satisfy th quation + lfl = 1&, th ara of th quadrilatral ABCD is givn by " (9) whr i=4 s = L1i. i=l (1) To driv (9), w mploy (7) in (1) and (), yilding and A =1{ in, (11),, (1)

5 Volum 9 No., Dmbr (13) I... and (9) follows immdiatly from (11) aftr th rplamnt of sin by its xprssion in (13). Th abov rsults ar summarizd in th following proposition: Proposition 1. A quadrilatral with dsignatd sid lngths 1,1, 1 3 and 1 4 ahivs th maximum ara if and only if it is a yli quadrilatral, and in this as th maximum ara A is givn by i=4 whr s = f). i=1 Rmark 1 It is wll known (s [1], for xampl) that if a irl q irumsribs a triangl, ABC whos thr sids ar of lngths 1 1, 1 and 1 3, thn its radius r 3 is givn by "'?_ ; * ' "' :>- E 'I,' )(:!;.... :u U) s::... n.!:),. o <( :s o Now suppos a irl C 4 irumsribs a quadrilatral whos four sids ar 1 1, 1, 1 3 and 1 4, thn w show that th radius r4 of 4 is givn by 1 osa> =- 1 1" 1 r' os =!_ (s Figur 3). r But os = os( 1 + ) =os 1 os -sin 1 sin, so I /-1 4 D (4r -1/)(4r -1/) ( ) = 4r - 4r A Figur 3 t;m.-t)' li-r 8 (') < > M E D L E y

6 It follows that (4r -z1)(4r -1) = ltl( )-r(1/ +1:/ -1/ -1/). ( ) On squaring both sids of (14) and rarranging trms, w obtain and hn ( Z 3 ZJ(z 1 Z XZ 1 Z 4 + Z)J r 4 =r= 4(s-1 1 )(s-l )(s-l 3 )(s-1 4 ). 3. Conditions For An n-gon To Ahiv Th Maximum Ara In this stion, w onsidr th st of n-gons whos sids hav fixd lngths lp1,...,ln. Using th rsults obtaind in Stion, w shall prov that:(1) An n-gon ahivs maximum ara if and only if its vrtis li on a irl; () Th valu of th maximum ara is indpndnt of th ordr in whih th n sids of th polygon ar arrangd. Proposition. An n-gon, with givn sid lngths lpl,...,1n, ahivs maximum ara if and only if it is yli. (14)... 3 : 3 :J> (a - 3 (lj Proof For th nssary part, onsidr a polygon A 1... An (s Figur 4) whos sids A 1, A 3,..., AnA 1 hav lngths lpl,...,ln rsptivly, and suppos that it ahivs th maximum ara. Assum that in this shap, A 1 A 4 has lngth 1. Considr th quadrilatral A 1 A A 4 If th n-gon An has ahivd maximum ara, thn so has th quadrilatral A 1 A 4 with A 1 A 4 = 1. From Stion, it follows that vrtis A 1,, and A 4 must li on th irumfrn of a An irl, say C. Similarly, th vrtis A,, A 4, and A 5 of th quadrilatral A 3 A 4 A 5 must also li on th irumfrn of a irl, say D. Howvr, as vry thr points dtrmin a irl, C and D must b th sam sin thy both ontain th points, A 3, and A 4 Indutivly, it follows that vry vrtx A; of th n-gon must li on C. Figur 4 \ f.mtt)' (..,.; < > ==== MEDLEY

7 = " Volum 9 No., Dmbr To prov th suffiiny, lt Q b an n-gon with sid lngths lpl,,ln whos vrtis Al'Az,...,An li on a irl C with ntr and radius r. Intuitivly, thr xists an n-gon that ahivs th maximum ara. Lt P b suh a polygon whos n vrtis ar BP B,..., Bn. Thn, by th nssity part, P is irumsribd by a irl C. Suppos th radius of C is rn and th ntr is ' (s Figur 5). If w an show that rn = r, thn th n-gon Q is ongrunt to th n-gon P, thus it also ahivs th maximum ara.. ' Figur 5 ' " " f/')... «' <t: ::3 - :E Suppos rn '1:- r, with no loss of gnrality, w assum that rn > r. Thn it follows that for ah i (i = 1,,,n), LA;A;+1 + < LB;B;+JBi+, using th onvntion that n + 1 = 1, n+=. Thus i=n i=n ILAiAi+1+ < ILBiBi+1Bi+. (15) i=1 i=1 Howvr, bothp andq ar n-gons, so i=n LLA;A;+ 1 Ai+ = (n- )1C and LLB;Bi+ 1 Bi+ = (n- )1C i=n 1 1 hold, whih ontradits th inquality (15). :u E (/) t:: Rmark From Proposition, on an also obsrv that th maximum ara ahivd by an n-gon with all its sid lngths spifid is indpndnt of th ordr in whih its n sids ar arrangd. As a mattr of fat, from Proposition, if an n-gon with givn sid lngths ahivs th maximum ara, thn it must b yli and hn an b thought of as th sum of n isosls triangls with th ntr of th irl as on ommon vrtx. It is thn obvious that th ara of th n-gon rmains unhangd rgardlss of how th n sids ar arrangd. r ti'

8 === M E D J " ' ' ;. 4. Summary And Rmarks ::s C/) 3 (\) In this papr, w drivd th formula of th maximum ara ahivd by a quadrilatral whos four sids ar prsribd by using lmntary alulus thniqus. W also provd that anngon ahivs th maximum ara if and only if it is yli. Th maximum ara of a quadrilatral with its four sids prsribd an also b obtaind using Brahmagupta's formula, whih stats that th ara of a quadrilatral quals (s- a)(s-b)(s- )(s-d)- abd os (AB) whr a, b,, and dar th sid lngths of th quadrilatral, s = t<a + b + +d), and A, B ar th angls btwn sids a and d, and sids band, rsptivly. Th intrstd radrs may lik to driv this formula from first prinipl. For n 5, w do not know of any losd form formula for th maximum ara ahivabl by an n-gon with prsribd sid lngths. Intrstd radrs my lik to rad mor about aras of yli polygons in Robbins (1995). REFERENCES [1] Portr, R.I., 197, Furthr Mathmatis, London: G. Bll & Sons, Ltd.. [] Robbins, D.P., 1995, Aras of Polygons Insribd in a Cirl, Amrian Mathmatial Monthly, 1, pp ",,rl