Cyclic Averages of Regular Polygons


 Mamuka Meskhishvili
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1 Cyclic Averages of Regular Polygos Maua Meshishvili Abstract We cosider a regular plae polygo with vertices ad a arbitrary poit i the plae Let R be the circuscribed radius of the polygo ad L a distace fro the poit to the cetroid of the polygo The the averages of the th powers of distaces fro the poit to the polygo vertices satisfy the relatios = R L, S = R L where =,, R L RL, Itroductio I his boo Matheatical Circus, Marti Garder wrote p 65: There is a beautifully syetric equatio for fidig the side of a equilateral triagle whe give the distaces of a poit fro its three corers: a 4 b 4 c 4 d 4 = a b c d This result was geeralized by J Beti [] fro a equilateral triagle to a regular polygo Cosider a regular plae polygo with vertices ad a arbitrary poit i the plae Deote by s ad q 4 respectively the averages of the squares ad the averages of the fourth powers of the distaces fro the poit to the vertices of the polygo s = d i ad q 4 = d 4 i The i= q 4 R 4 = s R is satisfied, where R is the circuscribed radius of the polygo Keywords ad phrases: Cetroid, regular polygo, average of distaces, polygoal, siplicial, polytopic distaces 00Matheatics Subject Classificatio: 5M04, 5N0, 5N5 i=
2 This result was geeralized to regular siplicial ad regular polytopic distaces i [] ad [], respectively I the aboveetioed papers the distaces are cosidered to the secod ad fourth powers oly Naturally, we are iterested to ow what happes if we cosider the distaces to higher ore tha 4 powers I the preset paper, for a regular polygo we itroduce a special id of averages of the distaces to the eve powers the cyclic averages ad by usig their properties establish the etrical relatios for regular polygos Geeral case Let us cosider a regular plae sided polygo A A A with the circuscribed radius R ad a arbitrary poit P i the plae Deote the distaces betwee P ad the cetroid O of the polygo by L, ad the distaces betwee P ad the ivertex by d i Fig Figure We use the followig otatio for the average of the th powers of the distaces: S = d i Theore For a regular polygo with vertices ad a arbitrary poit i the plae, let S be the averages of the th powers of the distaces fro the poit to the vertices If R is the circuscribed radius ad L the distace betwee the poit ad the cetroid, the = R L, S = R L where =,, i= R L RL,
3 First we eed to prove two leas Lea For arbitrary positive itegers ad, such that <, the followig coditio cos α π = 0 is satisfied, where α is a arbitrary agle Deote T = e iα e iα π e iα π e iα π The real part of T is ReT = cos α π The forula of the su of geoetric progressio gives T = e iα e i π e i π e i π = π iα e i = e, e i π e i π = cos π i π = Sice <, e i π So T = 0, ie ReT = 0, which proves Lea Rear If, the su always cotais α Lea For arbitrary positive itegers ad, such that < ad for a arbitrary agle α the followig coditios are satisfied: if is odd if is eve cos α π = 0; cos α π = Whe is odd, usig the powerreductio forula for cosie cos θ = =0 cos θ,
4 4 we obtai cos α π = = cos α cos α π = = [ 0 cos α 0 cos = cos α π cos α α π cos cos α π 0 cos α π α π cos cos α π [ cos α cos α π 0 cos α α π ] = cos α π cos αcos α π cos α cos cos α π α π cos α π ] Sice <, fro Lea it follows that each su equals zero, which proves the first part of Lea Whe is eve, the powerreductio forula for cosie is cos θ = =0 cos θ Aalogously to the case with odd, the su of the secod addeda vaishes, ad sice the uber of the first addeda is, the total su equals,
5 5 which proves Lea Proof of the theore We itroduce the ew otatios A = R L ad B = RL The π S = A B cos α A B cos α A B cos π α A B cos π α If =, by Lea we have π S = A B cos α A B cos α A B cos π α = A Therefore If >, we have S = R L π = A A B cos α cos α cos π α A B cos α cos π α ± cos π α A B cos α cos π α B cos π α cos α cos π α cos π α Accordig to Lea, all sus with the egative sig vaishes because they cotai odd powers ad there reai oly the sus with eve powers
6 6 If is eve, we have S = A = If is odd, we write A A B cos α cos π α B cos π α cos α cos π α cos π α = A B S = A A B cos α cos π α cos π α AB cos α cos π α = A cos π α = A B Usig the floor fuctio the iteger part, the obtaied results ca be cobied ito a sigle forula as follows S = A which proves the theore A B, The values of the averages S, S 4,, S reai costat whe the poit P oves o the circle CO, L, ie if we cosider ay poit o the circle P Fig, these averages will retai the sae values So we ca forulate
7 Defiitio The cyclic average of a regular polygo is the average of the power of the distaces fro the poit to the vertices, the value of which is costat for ay poit o the circle CO, L, where O is the cetroid of the polygo ad L is the distace betwee the poit ad the cetroid The properties of the cyclic average are as follows: Property Each regular go has a uber of cyclic averages, S 4,, S Property Cyclic averages ca be expressed oly i ters of the circuscribed radius R ad the distace L Property The expressios of the ocyclic averages cotai α, ie deped o the directio OP Fig Property 4 For fixed R ad L, the cyclic averages of equal powers of differet regular gos are the sae: = 4 = 5 = 6 =, S 4 = S 4 4 = S 4 5 = S 4 6 =, S 6 4 = S 6 5 = S 6 6 =, S 8 5 = S 8 6 = Property 5 Ay relatios i ters of the cyclic averages S, the circuscibed radius R ad the distace L, which are satisfied for a regular go, are at the sae tie satisfied for ay regular go, where, ie S ca be replaced by S 7 Equilateral triagle There are cyclic averages: Special cases = d d d = R L, S 4 = d4 d 4 d 4 = R L R L By eliiatig L, we obtai the forula itroduced by Garde d 4 d4 d4 d R 4 = d d R I ters of the cyclic averages S 4 R 4 = R By Property 5, for ay S 4 ad S, where Beti s result we have S 4 R 4 = R
8 8 Square There are cyclic averages: 4 = 4 d d d d 4 = R L, S 4 4 = 4 d4 d 4 d 4 d 4 4 = R L R L, S 6 4 = 4 d6 d 6 d 6 d 6 4 = R L 6R L R L Eliiatig L fro the cyclic averages 4 ad S 6 4 we obtai the secod relatio betwee the distaces ad the circuscribed radius Propositio For ay regular go, where 4, we have S 6 = S S R 5R 4 Substitutig R L = 4 ad R L = S ito S 6 4, we establish the direct correspodece betwee the distaces Propositio For ay regular go, where 4, S 6 = S S 4 S For the square, fro Propositio it follows that 8d 6 d 6 d 6 d 6 4d d d d 4 = 6d d d d 4d 4 d 4 d 4 d 4 4, which is equivalet to d d d d 4d d d d 4d d 4 d d = 0 Euerate the vertices of the square: A A A A 4 The oly d d = d d 4 holds, which together with the cyclic averages 4 ad S 4 4 iplies d d = d d 4 = R L, d d d d 4 = R 4 L 4 Refereces [] Beti, J, Regular polygoal distaces, Math Gaz, 8 July 997, pp [] Beti, J, Regular siplicial distaces, Math Gaz 79 July 995, p 06 [] Par, PS, Regular polytopic distaces, Foru Geo 6 06, 7 DEPARTMENT OF MATHEMATICS GEORGIANAMERICAN HIGH SCHOOL 8 CHKONDIDELI STR, TBILISI 080, GEORGIA Eail address: athaua@gailco
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