1.Introduction

Ronald van Luijks[1] showed that there exist arbitrarily many Heron triangles with all the same area and the same perimeter using elliptic K3 surface. 
Andrew Bremner[2] showed that how to find a set of n Heron triangles in two parameters such that
all triangles have equal perimeter and area for an arbitrary integer n > 2.

We show that there are Heron triangles such that all triangles have same perimeter and area for arbitrary n>1 using elementary method.
Furthermore, we show {7,9,50} Heron triangles with sides a,b,c has same area and the same perimeter. 


2. Theorem

There are n Heron triangles such that all triangles have same perimeter and area for arbitrary n>1.


Proof.

2s=a+b+c: perimeter
A: area
Heron triangle is given below by Heron's formula.

A^2=s(s-a)(s-b)(s-c).............................................................................(1)

Let assume there is a Heron triangle with sides a0,b0,c0 of perimeter 2s and area A.

Substitute a=a0+t, b=b0+kt to equation (1), we obtain

(sk+sk^2)t^2+((sa0-s^2)k^2+(2sa0-3s^2+2sb0)k-s^2+sb0)t
+(sa0^2+2s^3-2s^2b0-3s^2a0+2sb0a0)k+sb0^2+2s^3+2sb0a0-3s^2b0-2s^2a0=0............................(2)

Since equation (2) is a quadratic equation in t, for t to be rational number, the discriminant of the equation must be square number.

Let U = k, m = s/b0, n = a0/b0 then we obtain

V^2 = (m-n)^2U^4+(-2m^2+2mn+4m-4n)U^3+(6m+6mn-5m^2-6n)U^2+(-4n-2m^2+4mn+2m)U+(m-1)^2.............(3)
    
Quartic equation (3) has a rational point Q(U,V)=( 0, m-1 ), then this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+(-2m+4n)YX+(-4m^3+4m^2n+12m^2-12mn-8m+8n)Y =
X^3+(-6m^2+10mn+6m-4n^2-6n)X^2+(8mn+8m^3n-16m^2n-4n^2-4n^2m^2+8n^2m-4m^2-4m^4+8m^3)X
+24m^6-72m^5-88m^5n+248m^4n+72m^4+120n^2m^4-232m^3n-24m^3-72n^3m^3-312m^3n^2+264n^2m^2
+72m^2n+16n^4m^2+168m^2n^3-120n^3m-72n^2m-32n^4m+24n^3+16n^4.....................................(4)

The point corresponding to point Q is P(X,Y)=( [6m^2-10mn-6m+4n^2+6n, 16m^3-48m^2n-24m^2+48n^2m+48mn+8m-16n^3-8n-24n^2] ).

Hence we obtain 2P(X,Y)=( (m-1)(2n-2m+1)/((n-m)(n+2-2m)), -(m-1)(n+1-m)(3n^2+3n-6mn+4m^2+3-6m)/((n-m)(n+2-2m)^2)).

This point P is of infinite order, and the multiples hP, h = 2, 3, ...give infinitely many points.

Thus we can obtain infinitely many parametric solutions for equation (1).

Accordingly, we can construct the Heron triangles such that all triangles have same perimeter and area for arbitrary n>1.

Alternative method:
If rank of elliptic curve (4) is geater than 0, then this elliptic curve has infinitely many rational points.
Hence we can construct the Heron triangles such that all triangles have same perimeter and area for arbitrary n>1.

Q.E.D.


3.Examples

7 Heron triangles with sides a,b,c has same area and the same perimeter. 

7 rational Heron triangles:
[a,b,c,s,A]=[5, 35/2, 39/2, 21, 42] ,[15, 20, 7, 21, 42], [1025/66, 2273/114, 1365/209, 21, 42],
            [7995/782, 49175/2438, 10457/901, 21, 42], [40740/2021, 3709/387, 5185/423, 21, 42],
            [24315/5341, 112628/5929, 243355/13189, 21, 42], [1216225/62823, 140605/29283, 1972068/110549, 21, 42]
             
            a                        b                      c
[ 5116557011014477336770, 17907949538550670678695, 19954572342956461613403]
[15349671033043432010310, 20466228044057909347080,  7163179815420268271478]
[15892336170575270515725, 20403393133396328046453,  6683349588550011066690]
[10462115934286635884265, 20640417638772512144025, 11876545319462461600578]
[20628256568899535546760,  9807395324988473613478, 12543426998633600468630]
[ 4658643839086950625110, 19438946973740548270728, 18881488079694110733030]
[19810848099330126542550,  4913523194574945093990, 18254707598616537992328]

2s= 42979078892521609628868
A=  43980981486895155962956221474129519349975272


9 Heron triangles with sides a,b,c has same area and the same perimeter. 

9 rational Heron triangles:
[a,b,c,s,A]=[13, 20, 21, 27, 126], [27/2, 113/6, 65/3, 27, 126], [87/5, 29/2, 221/10, 27, 126],
            [9828/725, 35881/1653, 26701/1425, 27, 126], [87845/6767, 29597/1474, 46527/2222, 27, 126],
            [23348/1593, 53811/3127, 31675/1431, 27, 126], [8827/551, 35484/1595, 16441/1045, 27, 126],
            [149019/6698, 55745/3502, 321281/20291, 27, 126], [16884543/964603, 18384313/832661, 3642380/252647, 27, 126]
             
             a                                b                               c
[349754510248707532407709777650, 538083861921088511396476581000, 564988055017142936966300410050]
[363206606796734745192621692175, 506695636642358348231682113775, 582924183747845887346182962750]
[468132959871347004914934625470, 390110799892789170762445521225, 594582667422802805093106622005]
[364709530686928130345142885384, 583998398354221926116665946850, 504118498145788924308677936466]
[349253560295980643443353666750, 540219405063719697143877794025, 563353461827238640183255307925]
[394324607913797067297078945800, 462980983272076972925420551650, 595520836001064940547987271250]
[431004196840058828502422756850, 598538174182065979259955329160, 423284056164814173008108682690]
[598572103759470654522182320275, 428262205636651614331761664875, 425992117790816711916542783550]
[470934680081478161351343448050, 594017381491751895081965683650, 387874365613708924337177637000]

2s= 1452826427186938980770486768700
A=  91203286374872609779271205350724129887502370303656611715000


50 Heron triangles with sides a,b,c has same area and the same perimeter. 

                                      a                                                                                            b                                                                               c
[11119332954600771473095652400116289969305881732148321366888686520426385341661392553600, 5815283326831437954320140048336680386246179526583317496476267088268971644202222542400, 7988026547845381805384807758704231299788708140911150407247619626743092917860195800000]
[11183237166983534527538730862185923819704191397275610570146667477440330085004274120000, 6646038087807357662080160055241920441424205173238077138830019529450253307659682905600, 7093367574486699043181709289729357394212372829129101561635886228547866511059853870400]
[10314139878577956987112863778038903454287179951544477405838126462050681575541084816960, 4473294866793413811015492344874369527881676558910244228058666990976132034001709648000,10135208083906220434672244084243928673171912889188067636715779782411636294181016431040]
[10753336101863128524921657571899296098842835468237483202774795584801065447970343582400, 4728911716324466028787806193152904929474915219419401041090590819031911007373235913600, 9440395011089996679091136442105000627023018711985905026747186831605473448380231400000]
[8286246205631609392785840581695855934980819911505166689118197426141501720126976443200, 5559666477300385736547826200058144984652940866074160683444343260213192670830696276800, 11076730146345596103466933425403200735707008622063461898050032549083755512766138176000]
[9479124836776519742389971873662354475749266993881231816600508623735136929194099016000, 4707610312196878343973446705796360312675478664376971306671263833360596092925608724800, 10735907680304193146437181627698486866916023741384586147340800778342716881604103155200]
[8147787078802289441492503913878315925784482303729373415392572019277954776217399716000, 11098031550473183788281292912759745352506445177105891632469359534755070427213765364800, 5676824200002118003026803380519140377049841918807524222750641681405424700292645815200]
[10241928734826634183644783409810169658822629068616649548193283563854982009244856640000, 4468461775100599798494587251104397219868359021211541683274449943807010162656449697600,10212252319350357250661229546242634776649781309814598039144839727776457731822504558400]
[9085511934418921218646372650769682208803156737662421506678162149373883075270553136000, 4949026225642872105202854229170532636402426288191174963423636337635498456665383531200, 10888104669215797908951373327216986810135186373789192800510774748429068371787874228800]
[10460978255532158482236124165575811505620101815984381245854990813400659767124040200000, 4508151709911284567984444233275987991735300012616038338926656603892829166734190502400, 9953512863834148182580031808305402157985367571042369685830925818144960969865580193600]
[11185549490457910822271868569695022412975182865421663797896133893648071506638391545100, 6788684951980646090109446653236366280858202908707970375823840798821938013387910798400, 6948408386839034320419284984225812961507383625513155096892598542968440383697508552500]
[11176568901343594034901192240056918548358280823523197783719747725404092372655451695680, 7291094725741601199870293237074224484268319204758243449363521424955719069873710952320, 6454979202192395998029114730026058622714169371361348037529304085078638461194648248000]
[9330015007883405948689455462166542158153211108584223675665219724035932528060708694400, 4792815928707229083230884655222538779873224884546690244348571776045855750716117480000, 10799811892686956200880260089768120717314333406511875350598781735356661624946984721600]
[10071045672685547226473840052630579130448761568848142921525443346783493794299375747200, 4482135370482728858072835215121156819438754971082003604003446965266598698100527256000,10369461786109315148253924939405465705453252859712642745083682923388357411323907892800]
[11093508649596778183971394665444314646199715497610581209407721613139928219351597948000, 8178723720796903774972207925515861956922513311477296709134175403125527477994624846912, 5650410458883909273856997616197025052218540590554911352070676219172994206377588101088]
[10925513585176127483795766075597400488976942528025993673490854167285190295918147363200, 5032879372049883943837158242891533667612906695342326933836225106627341294490963576000, 8964249872051579805167675888668267498750920176274468663285493961525918313314699956800]
[9584499696171456458804313195959591920664589952191519065804777747735165236294270522800, 4652904433414664517063841658721598001349652966199822216003446801977446426276020717200, 10685238699691470256932445352476011733326526481251447988804348685725838241153519656000]
[11135555482568906273732675123306477524596336219368572903760628390299490528961619150400, 5941347147214421492458103728314289731167415276495694883287511625310636148006026320000, 7845740199494263466609821355536434399577017903778521483564433219828323226756165425600]
[10696766350219750775590796069680836133547059071494121510345287442171675726306588195780, 4664463638304402863243084510129486961200875259269751929719775981532741202384883014720, 9561412840753437593966719627346878560592835068878915830547509811734032975032339685500]
[10833888358720098508066324995593583543917502603791420339581667103208059568140678691200, 4847094224559874544204435041054742713854536345305104431588263630709325738426063336000, 9241660245997618180529840170508875397568730450546264499442642501521064597157068868800]
[10040673921450339916203000428285746184134413419051491539482421746001352518684133010400, 4487702978338599115067645313776011112188987756617988478130095058207392785153287720000,10394265929488652201529954465095444359017368223973309253000056431229704599886390165600]
[8570694975430153565777590312795362852136692908529949265129802809523300090538872360000, 11023706783250094225360601616238980732185241879109706730800963515723045458615892106400, 5328241070597343441662408278122858071018834612003133274681806910192104354569046429600]
[11105089867868941391344442791462885338860926582824214275174954121480481845168888459200, 5719958952311533908624804362408169959014960669834064789285671742845284923832818740800, 8097594009097115932831353053286146357464882146984510206151947371112683134722103696000]
[11185514940890246874033731678655732551005517286329652561549922244224985541202634255040, 6952418283512886447043877076206756535446523624073700304872508512091757401095131992960, 6784709604874457911722991452294712568888728489239436404190142479121706961426044648000]
[9354849528221162468400700033250498225376469348744587753691223895373077881548829496000, 4778044366772853861864174720742482570874775508763331762402443617796567777731308899200, 10789748934283574902535725453164220859089524542134869754518905722268804244443672500800]
[10214398797925991590553972930847383499639930569522481347564414533637285603405298648000, 4468425352754849843843425283311147173193511185040483477738906644302750749166797528640,10239818678596749798403201992998670982507327645079824445309252057498413551151714719360]
[11053670936926159472418830844566371116771084429320606604151241221008956246788699291200, 5450232202395442662342977248099575388833425371881551265602980521986352835507791268800, 8418739689955989098038792114491255149736259598440631400858351492443140821427320336000]
[10849012777377878070364999220156114494167120040721512583086189505720123274826802866800, 4873467984190305643488151554815252999276888652732701376479719239910073452157314626000, 9200162067709407518947449432185834161896760706188575311046664489808253176739693403200]
[10153119149155077216623854323376520337553617383673024380544264989794216374130011496000, 4471581307272888356775884752237289623581462389231199986625076403074759449470772900800,10297942372849625659400861131543391694205689626738564903443231842569474080123026499200]
[11044562438897862474172374361500402589690045727201816995465380866268369022527413716800, 5410999505041941125483782958777645444952923162794666674201763861255595648376050803200, 8467080885337787633144442886879153620697800509646305600945428507914485232820346376000]
[11068488766370279875910100524889469108696489851230681789768582456515623678039085080000, 8335420090090169220637814906189948214395735425801623774073158230901843791969262134400, 5518733972817142136252684776077784332248544122610483706770832548020982433715463681600]
[8599989027422838562815519541368129653089823130568519595631598556740267578816734671200, 5305186966365689702169382491260270474724188250773053511404775044295592914248364788800, 11017466835489062967815698174528801527526758018301216163576199634402589410658711436000]
[11077353675367846069201227848767895922867775059752942295848235384495753092698831900800, 5562853974097818427937472980454601600402295489056792766422635514386880464127481496000, 8282435179811926735661899377934704132070698850833054208341702336555816346897497499200]
[11177937643951971936651738246775682344175680596353432524130722107374806255920393456000, 6485956296858303490923377305087410112555383516946056763138017975199935311629553332800, 7258748888467315805225484655294109198609705286343299983343833152863708336173864107200]
[11100758196102315716032461468979599857523359567677378746924929596052389608025739880000, 8128685067093654970372101471724448853705758192509807437416902214848440287915328254400, 5693199566081620546396037266453152944111651639455603086270741424537620007782742761600]
[10384865610289864637168948636793034025393613338484427079779454338076673226102495248000, 4485470221272656389015583014355145404787410567749734859520294156335362090510700822080,10052306997715070206616068556009022225159745493408627331312824741026414587110614825920]
[10938263613526467378328293737153653472381391316990140452219708185601264629924327828800, 5064460214237918601092756619390014953556649427518559371270701208317088986828051136000, 8919919001513205253379549850613533229402728655134089447122163841520096286971431931200]
[10983527365759891579999849189371648277164518547667106677100678030446396158199478376000, 5191394493885741603331149358374454266709541137726007906547897258360901034463654403200, 8747720969631958049469601659411099111466709714249674686963997946631152711060678116800]
[11181871349017093778323829763549786309700707283235927496906303288014585729162145648000, 7145545034494068464953789848450130599898415359830970684296010886233479126242907685440, 6595226445766428989522980595157284745741646756575891089410259061190385048318757562560]
[7238828784335553271765143489505195530688006356383759562330128849156553823368341067975, 6505087704507820643448384738741883249633565395115755271437357065904657729354434600000, 11178726340434217317587071978910122875019197648143274436845087320377238351001035228025]
[11135136958835852528946692346298612708608387855210690478475079465492532792272346248000, 7849705373760460827358415974281624107421965788461497697836077515364914888775157609280, 5937800496681277876495491886576964839310415755970601094301416254581002222676307038720]
[10567866187465328079083471761183791752756154289140777589128057633134707484086442629440, 4560328891353616685586805548394999816309987523550835570001143381057280456119012618560, 9794447750458646468130322897578410086274627586951176111483372221246461963518355648000]
[6756869722370527814109235173182599731181005101403470773786954063958218995079300208400, 11185225386400750143259880909896877359651692031329973094804678352141861767316377640000, 6980547720506313275431484124077724564508072266909345402020940819338369141328133047600]
[11112938036232841208252720832694543358801561088671121842115087018605566944403575693120, 8038720804541385413418981258897719640624760245241883448673470221345991809057149248000, 5770983988503364611128898115564938655914448065729783979824015995486891150263085954880]
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[9219241298849036619688019002170936033155061832388613894241383495532506945797851176000, 4861275111284409588534898836323119144186863504035229345170236066723364673447554158050, 10842126419144145024577682368663146477998844063218946031200953673182578284478405561950]
[9699747696503526685272195026772288757744014323290932768235338646984057138884193665600, 4599120226523194248158450555087794941350975266857974119147675221298299853700444510400, 10623774906250870299369954625297117956245779809493882383229559367156092911139172720000]
[11185695266516498221533005199721054008493608813532959380550112666173836955113238648000, 6928635256399909430484762323066401974305856588561509804583812398761251706072581984640, 6808312306361183580782832684369745672541303997548320085478648170503361242537990263360]
[10287877792408554115310090779934275600546769332448789027185240246588729785524502645600, 10164059590611007815313793791482351069638304807480258578955302388186521759981145230400,4470705446258029302176715635740574985155695259713741664472030600663198358218163020000]
[11170913696045332656863609310332829031822605958257829425975561154424832975228947995200, 7404326533549663794198697082407669600881504077928341259973927187307668414924782084800, 6347402599682594781738293814416703022636659363456618584663084893705948513570080816000]

2s= 24922642829277591232800600207157201655340769399642789270612573235438449903723810896000
A=  22297266047026626475657720384640652038455916045155383639549638449627502740893235068827321037981006453508750570786760233463330752407638659106674323828968813970144921600000

4.References

[1] Ronald van Luijks: An elliptic K3 surface associated to Heron triangles. Journal of Number
Theory, 123(1):92–119, mar 2007.
[2] Andrew Bremner: On heron triangles. In Annales Mathematicae et Informaticae, volume 33, 2006.