A Cornucopia of Pythagorean triangles

Transcription

1 A onucopi of Pytgoen tingles onstntine Zelto Deptment of temtics 0 ckey Hll 9 Univesity Plce Univesity of Pittsbug Pittsbug PA 60 USA Also: onstntine Zelto PO Bo 80 Pittsbug PA 0 USA e-mil ddesses: ) onstntine_zelto@yoocom ) kzet9@pittedu Pge of 6

2 Intoduction As wit mny woks in mtemtics te oigin of tis ticle lies in discussion wit collegue bout clssoom question In Figue two cicles e illustted centeed t e two cicles ve only one point of intesection I ; tt point I being te point of tngency between te two cicles; tey two cicles being etenlly tngentil e two cicles ve tee tngents o tngent lines in common ese e te two conguent tngents s well s tei tid common tngent wic is te line pependicul to midpoint of te line segment t I ; wic psses toug te e foementioned subject of discussion ws simply te clcultion of te lengt of te tngent in tems of te two dii I I is s it tuns out is simple mtte te nswe (s we will see below) is is ten led to numbe teoetic eplotion by tis uto wic esulted in tis wok If we look t Figue we cn identify siteen igt tingles wic e listed in Section Now conside te cse wen te two dii e integes In Section 6 we give te pecise conditions te dii must stisfy in ode tt ll te siteen igt tingles ( listed in Section ) e ctully Pytgoen In Section we offe some immedite geometic obsevtions fom Figue in Section we compute te side lengts of tese Pytgoen tingles In Section we stte tee esults fom numbe teoy ( we offe poof fo esult ) ; including te well known pmetic fomuls wic descibe te entie fmily of Pytgoen tiples ( esult ) In Section we list te sidelengts of te 6 Pytgoen tingles in Section 8 we pesent numeicl emple Finlly in Section 9 we eplin wy te digonl lengts d d ; e lwys itionl numbes wen te 6 igt tingles e Pytgoen tis immeditely follows fom well known eisting numbe teoy esults Nottion: ) If X Y e two points on te plne we will denote by XY te stigt line segment joining te two points by XY te lengt of te line segment XY ) We will denote by XY te full stigt line tt goes toug te two points X Y ; by YX te lf-line o y wic emntes fom Y ( o wose vete is te point Y ) wic only contins tose points on ( of te full line XY ) wic lie on te side of (te point Y ) wic contins X ) If X Y Z e tee points on te plne we will denote by XYZ te ngle fomed by te ys YX YZ ) If b e ntul numbes (o positive integes) gcd b will denote te getest common diviso of b Pge of 6

3 I Figue 90 I I Also fom te igt tingle I since is te midpoint of ; we ve I e siteen igt tingles ese e (fom Figue ) : e two conguent igt tingles I e two conguent igt tingles I e fou conguent igt tingles I I e two conguent igt tingles I e two conguent igt tingles I e igt tingle I e igt tingle I e igt tingle e igt tingle I e igt tingle e igt tingle Pge of 6

4 Immedite geometic obsevtions Looking t Figue we see tt te geomety involved is petty obvious esy Fist conside te vious ngels We ve I I And I I I I I I I I I I And wit 90 Also te lines I e pependicul; so e te lines e pependicul Likewise te lines I I ; te lines pependicul s well And bot lines e pependicul to te line I e Lengt omputtions In tis section we compute te side lengts of te 6 igt tingles (listed in Section ) in tems of te dii e key clcultions e tose of te lengts I I e est of te lengts follow esily fom tese five lengts Let F be te foot of te pependicul fom te point to te line segment (Figue ) en F F F Figue And fom te igt tingle F we ve F F F ; by vitue of F we obtin Pge of 6

5 ; o equivlently ; () Fom te igt tingle I I I ; I we get ; by () ; since we obtin () Woking similly fom te igt tingle I we get (b) Net fom te igt tingles we get sin since I I I I I I I I So tt sin sin I sin I () Howeve 90 so sin cos cos sin eefoe fom te identity sin cos () we obtin () oeove fom te igt tingle I we ve I I ; by () () By solving te line system of equtions () () in Pge of 6 (my use me s ule)

6 Pge 6 of 6 We fute obtin Likewise Hence I (6) I (6b) And I (6c) I (6d) Net we compute te eigt Fom te igt tingle I we ve ; ; ) (6 ; ; by I I Also Similly we clculte in tems of fom tingle I

7 Altogete we ve () ( ) (b) (c) (d) Futemoe fom te igt tingle I we ve I (8) I ; ny by () emk: Note tt I I s it cn be esily seen fom (b) (d) (6c) (6d) is cn lso be seen fom te fct tt te qudiltel I is ectngle o finis te lengt computtions; we must compute te lengts Fom te simility of te igt tingles we obtin ( ) ; solving fo yields tus (9) o equivlently (9b) We could compute fom te igt tingles o ltentively by using te simility of te tingles once moe: we obtin fte solving fo ; by () Pge of 6

8 (9c) tus o equivlently (9d) ee esults fom numbe teoy e pmetic fomuls listed below in esults descibe te entie fmily of Pytgoen tiples A welt of istoicl infomtion on Pytgoen tingles cn be found in efeences [] [] esult : A tiple b cof positive integes b c is Pytgoen one wit ypotenuse lengt c (ie b c ) if only if ( b my be switced) m n b kmn c km n k fo some positive integes k m n suc tt m n e eltively pime m n m n ve diffeent pities (ie one of tem is even te ote odd) Wen k te Pytgoen tiple is clled pimitive Also note tt k gcd( b) gcd( c) gcd( b c) e following esult is well known it cn esily be found in numbe teoy books; fo emple see [] esult : Suppose tt ntul numbe en c c c b e eltively pime positive integes suc tt c esult : Suppose tt b e positive integes suc tt en pime c Poof : Suppose tt b Pge 8 of 6 b c wee c is b c wee c c e eltively pime integes suc tt b c fo some ntul numbe c fo some positive integes ; suc tt e eltively b c Let be te getest common of b ; gcd b c b c fo eltively pime positive integes c c We obtin cc is diviso of en c Since n n c it follows tt is diviso of c ( moe genelly if is diviso of c ; ten must be diviso of c - tis is typiclly n eecise in elementy numbe teoy couse) Put c c We ve cc c ; cc c since gcd c c it follows fom esult tt

9 c c And so c ; c c wit being eltively pime Altogete we ve b Befoe we poceed fute note tt by inspection we see fom Figue becuse of te ectngle I tt ; wee ( fom Section ) I I I 6 e integlity of te lengts ; te tionlity of te lengts Wen e integes it is cle fom () tt will be n intege pecisely wen te poduct is pefect o intege sque Otewise te lengt will be n itionl numbe By esult will be n intege sque if only if wee e ntul numbes wit being eltively pime Now if in ddition to te lengt being n intege; we lso equie te lengts to be integes; it becomes ppent fom fomuls () (b) tt since ; will be integes if only if fo some ntul numbe In ote wods pecisely wen is pimitive Pytgoen tiple Summy Wen te two dii e integes ten te tee lengts will be integes if only if wee e positive integes suc tt e eltively pime fo some positive intege en is pimitive Pytgoen tiple so by esult Eite m n mn ; o ltentively mn m n And in eite cse m n ; wee m n e eltively pime ntul numbes suc tt m n m n mod (ie one of m n is even te ote odd) And wit since Obviously by (8) (0) te lengts I e lso integes Futemoe by (0) (6)-(6d) ()-(d) (9)-(9d) te ten lengts Pge 9 of 6

10 Pge 0 of 6 ; e tionl numbes Net we clculte te bove lengts in tems of te integes e computtions e stigtfowd One simply uses (0) in conjunction wit (6)-(6d) ()-(d) (8) (9)-(9d) in ode to obtin te following: () (b) (c) (d) (e) (f) (g) ; () Similly (i) I (j) (k) (l) (m)

11 (n) Now since is pimitive Pytgoen tiple; te tee integes e piwise eltively pime; is odd one of is odd te ote even; teefoe e following copimeness conditions follow edily (fom (0) ) ( my lso use In (d) gcd In (e) gcd In (f) gcd In(g) gcd In() gcd In (i) gcd () In (k) gcd In (l) gcd In (m) gcd In (n) gcd ): is lso odd A fundmentl esult in numbe teoy is Euclid s Lemm wic sys tt if n intege divides te poduct bc is eltively pime to b ten must be diviso of te intege c Applying Euclid s Lemm to te fomuls in (d)-(i) (k)-(n) in conjunction wit te copimeness conditions () te following becomes cle: In ode tt te lengts ; be integes s well it is necessy sufficient tt te intege be divisible by bot divisible by te lest common multible of of ; ie must be e lest common multiple is tei poduct in vitue of te fct tt gcd Pge of 6

12 en we must ve t ( dii) wee t is positive intege t t () onsequently fom ()-(n) () we obtin te following lengt fomuls: t () t (b) t (c) t (d) t (e) t (f) t (g) t () t (i) t I (j) (k) t t (l) (m) t t (n) Pge of 6

13 e siteen Pytgoen tingles Below we list te siteen Pytgoen tingles obtined wen ll te lengts I e integes ese lengts e clculted in tems of te integes t s epessed in te fomuls ()-(n) ese 6 Pytgoen tingles e: ) e two conguent Pytgoen tingles I ey ve ypotenuse lengt I leg lengts I ) e two conguent Pytgoen tingles I ey ve ypotenuse lengt I leg lengts I ) e fou conguent Pytgoen tingles I I e ve ypotenuse lengt I I ; leg lengts lso I ) e two conguent Pytgoen tingles I ey ve ypotenuse lengt leg lengts I I ) e two conguent Pytgoen tingles I ey ve ypotenuse lengt leg lengts I I 6) e Pytgoen tingle It s ypotenuse lengt leg lengts ) e Pytgoen tingle I It s ypotenuse lengt leg lengts I I 8) e Pytgoen tingle It s ypotenuse lengt leg lengts Pge of 6

14 Pge of 6 9) e Pytgoen tingle It s ypotenuse lengt leg lengts 8 A numeicl emple e fist pimitive Pytgoen tiple wit is We tke t we pply fomuls () ()-(n); in ode to compute te numeicl vlues of te vious lengts Specificlly: I 9 e itionlity of te digonl lengts d d ee e two igt tingles in Figue tt we ve not mentioned tusf ese e te tingles As we ve seen insof unde te conditions (0) (); te esulting fomuls ()-(n); siteen Pytgoen tingles listed in Section e fomed in Figue e tingles ve integl leg lengts; tese being Howeve s we will see below te two ypotenuse lengts d d e bot itionl numbes (qudtic itionls) Let us see wy We ve d ; by(0) () we obtin

15 d d d ; ; () And similly d (b) We point out tt we do not need to mke use of te specil fom tt te intege s unde () In ote wods we will pove tt bot d d e itionl unde (0); so wen only te lengts I e integl fo sue te est of te lengts e tionl genelly speking ccoding to ()-(n); te two lengts d d must be itionl numbes Indeed conside () (0) We know tt eite m n mn o vice-ves lely d will be eite n intege o n itionl numbe depending on wete sque o not We ve is n intege m n mn m m n n ; ecll fom (0) tt m n e eltively pime integes one being even te ote odd n it be m m n n l fo some positive intege l? e nswe is no In efeence [] Eule is mentioned s ving poved tt te diopntine eqution y y z wit gcd y y e odd cn only ve positive intege solutions wen bot In efeence [] poof cn be found of te fct tt te diopntine eqution y y z s no solutions in positive integes y z suc tt gcd y y mod (ie one of y is odd te ote even)net suppose tt mn We ve mn m n m m n n m n eefoe will be pefect sque pecisely wen impossible since te diopntine eqution integes suc tt gcd y y z m y m n n is pefect sque; wic is y z s unique solution in positive t solution is y z (ecll tt m n bove ve diffeent pities) A poof of tis esult cn be found in [] Fo te oiginl ppe tt estblised te sid esult efe to [] It is obvious fom (b) tt te itionlity of d is estblised in n identicl mnne Pge of 6

16 efeences [] Dikson LE Histoy of eoy of Numbes VolII AS else Publising Povidence ode Isl 99 ISBN: ; 80pp (unlteed tetul epint of te oiginl book fist publised by negie Institute of Wsington in ) () Fo mteil on Pytgoen tingles tionl igt tingles see pges 6-90 (b) Fo Eule s mention in eltion to te diopntine eqution pge 6 y y z [] Siepinski W Elementy eoy of Numbes oiginl edition Wsw Pol pp (no ISBN numbe)oe ecent vesion (988) publised by Elsevie Publising distibuted by Not-Holl Not-Holl temticl Liby Amstedm (988) is book is vilble by vious libies but it is only pinted upon dem Specificlly UI Books on Dem Fom: Po Quest ompny 00 Not Zeeb od Ann Abo icign USA; ISBN: () Fo desciption deivtion of Pytgoen tiples see pges 8- see (b) Fo te diopntie eqution y y z see pges - [] Zelto onstntine e Seventeen Elements of Pytgoen tingles publised in ivog Xiv: 0800 July 008 [] Pocklington H Some Diopntine impossibilities Poc mbpilsoc (9)pp0-8 Pge 6 of 6