# Cyclic Averages of Regular Polygons

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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=

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 power-reductio 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 power-reductio 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 o-cyclic 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, P-S, Regular polytopic distaces, Foru Geo 6 06, 7 DEPARTMENT OF MATHEMATICS GEORGIAN-AMERICAN HIGH SCHOOL 8 CHKONDIDELI STR, TBILISI 080, GEORGIA E-ail address: athaua@gailco

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