Algebraic Methods in Plane Geometry

Algebraic Methods in Plane Geometry 1. The Use of Conic Sections Shailesh A Shirali Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject. Written with affection and respect for Professor A R Rao of Ahmedabad, mathematician, teacher and a continuing source of inspiration to a vast number of students, on the occasion of his one hundredth birthday. May there be many more! Keywords. Conics, family of curves, Pascal s theorem, homogeneous coordinates, Butterfly theorem, abelian group, associativity of addition, group law. `R id e r s ' in g e o m e tr y a r e a lw a y s a p le a s u r e t o t a c k le, a n d t h is p le a s u r e is d o u b le d w h e n o n e n d s c o n n e c t io n s b e tw e e n p la n e g e o m e t r y a n d a lg e b r a. T h is t h r e e -p a r t a r t ic le is a b o u t s u c h c o n n e c t io n s. I n P a r t s 1 a n d 2, w e e x p lo r e s o m e c o n n e c t io n s b e t w e e n p la n e g e o m e t r y a n d t h e a l- g e b r a o f c o n ic s a n d c u b ic s ; in P a r t 1 w e g iv e a lg e b r a ic p r o o fs o f r e s u lt s s u c h a s P a s c a l's T h e - o r e m a n d t h e B u t t e r y T h e o r e m, a n d in P a r t 2 w e s t u d y s o m e g r o u p t h e o r e t ic a n d n u m b e r t h e o - r e t ic a s p e c ts o f c u b ic c u r v e s. In P a r t 3 w e lo o k a t t h e r o le o f m a p p in g s a n d t r a n s fo r m a t io n g r o u p s in p la n e g e o m e t r y. 1. P a r a b o la in a T r ia n g le W e rst re ca ll tw o re su lts fro m th e g eo m e try o f th e p a ra b o la. L e t P d e n o te a p a ra b o la w ith fo c u s F a n d d ire c trix `. F o r a n y p o in t P 2 P, le t t P d e n o te th e ta n g en t to P a t P. (i) T h e im a g e o f F u n d e r re e ctio n in a n y o f th e ta n g e n ts t P lie s o n th e d ire c trix `. (S e e F ig u re 1 a ). C o n v erse ly, if th e im a g e o f F u n d e r re e c tio n in a lin e m lie s o n `, th e n m is ta n g e n t to P. (T h e c o lle c tio n o f a ll su ch lin e s m e n v e lo p e s th e p a ra b o la in a v isu a lly v ery a ttra c tiv e w a y, a s c a n b e sh o w n u sin g p a p er fo ld in g.) (ii) If A ; B ; C a re th re e d istin c t p o in ts o n P, th e n th e circ u m c ircle o f th e tria n g le P Q R w h o se sid e s lie o n th e ta n g en ts t A ; t B ; t C, re sp e c tiv ely, p a sse s th ro u g h th e fo c u s F. (S e e F ig u re 1 b ). 916 RESONANCE October 2008

(a) (b) S o m e re a d e rs m a y re co g n iz e th a t th e se tw o re su lts co m e to g eth e r in th e W a lla ce{ S im so n th eo rem : \ F o r a n y tria n g le th e fe e t o f th e n o rm a ls fro m a p o in t o n its c irc u m - circ le to th e th re e sid es o f th e tria n g le lie in a stra ig h t lin e ". T h e re su lt b e lo w, w h o se so u rce is a p ro b le m fro m th e p ro b le m so lv in g m a g a zin e C ru x M a th em a tico ru m, b rin g s th ese tw o resu lts in a p retty w a y. T h e o r e m 1. In tria n gle A B C let the feet o f the a ltitu d es fro m A ; B ; C be D ; E ; F, resp ectiv ely. L e t à E! F c u t Ã! A D in K, let L be th e m id po in t o f K D, a n d let th e n o r- m a l to A D a t L cu t à A! C in Q, a n d à A! B in R. T h en th e po in ts A ; R ; D ; Q a re co n cy clic. (S e e F igu re 2.) P roo f. C o n sid e r th e p a ra b o la P w ith fo cu s D, a n d d i- rec trix à E! F (F igu re 3 ). O b se rv e th a t: Figure 1. For any triangle the feet of the normals from a point on its circumcircle to the three sides of the triangle lie in a straight line. Figure 2. RESONANCE October 2008 917

Figure 3. ² à Q! R is ta n ge n t to P. T h is is b ec a u se th e im a g e o f D u n d e r re e ctio n in à Q! R lie s o n th e d ire ctrix à E! F. (T h e im a g e is K.) ² à A! B is ta n g en t to P. T h is is b e c a u se th e im a g e o f D u n d er re ec tio n in à A! B lie s o n th e d irec trix. In tu rn th is is tru e b ec a u se o f th e k n o w n p ro p e rty th a t th e a ltitu d e s A D, B E, C F b ise c t th e a n g les o f th e o rth ic tria n g le 4 D E F. H en c e, F E ; F D m a k e e q u a l a n g le s w ith à A! B, im p ly in g th a t th e im a g e o f D u n d e r re ec tio n in à A! B lies o n à E! F, a s cla im ed. It is a known ² à A! C is ta n gen t to P. T h is is so b ec a u se th e im a g e property of a triangle o f D u n d er re e ctio n in à A! C lies o n th e d ire ctrix (a s formed by three a b o v e). tangents to a ² T h e circu m circle o f 4 A R Q pa sses th ro u gh D. F o r, parabola that its th e ta n g e n ts à R! Q, à A! B, à A! C a re th e sid es o f 4 A R Q, circumcircle passes a n d it is a k n o w n p ro p erty o f a tria n g le fo rm e d b y through the focus of th re e ta n g en ts to a p a ra b o la th a t its c irc u m circ le the parabola. p a sse s th ro u g h th e fo c u s o f th e p a ra b o la. 918 RESONANCE October 2008

N o te th a t sin c e à R! Q is p a ra llel to à B! C, th e circu m circles o f 4 A R Q a n d 4 A B C w ill b e ta n g en t to e a ch o th e r a t A, a s F igu re 3 sh o w s. 2. F a m ilie s o f C u r v e s W e n o w sta te a n e x tre m ely u se fu l re su lt c o n c e rn in g fa m - ilies o f cu rves w ith a given d egree. T h e e n tire d iscu ssio n is w ith re fere n c e to a x e d re cta n g u la r c o o rd in a te sy s- tem o n a g iv e n p la n e. C o n sid e r th e fa m ily L o f a ll p o ssib le stra ig h t lin e s. T h e g e n e ra l eq u a tio n o f a stra ig h t lin e is a x + b y + c = 0 (w ith a ; b n o t b o th ze ro ). T h ere a re th re e c o e ± cie n ts in th is eq u a tio n, b u t if w e m u ltip ly a ll o f th em b y a n o n -z e ro co n sta n t w e g e t th e sa m e lin e. H en c e, L h a s `tw o d e g re e s o f free d o m ' (to b o rro w a n e x p ressio n fro m p h y sic s); o r p h ra sed o th e rw ise, L is a tw o p a ra m ete r fa m ily, a n d tw o d istin c t p o in ts a re n e e d ed to x a stra ig h t lin e. If w e x o n e p o in t a n d a llo w th e o th er o n e to v a ry, th e n w e g e t th e o n e p a ra m e ter fa m ily o f a ll lin e s p a ssin g th ro u g h a p o in t, a lso c a lle d a `p e n cil o f lin e s'. T h e te rm `p en cil' m a y b e u n d e rsto o d fro m F igu re 4 a. N e x t, co n sid er th e fa m ily F o f a ll p o ssib le sec o n d d eg ree cu rv e s, i.e., a ll p o ssib le c u rv e s o f th e ty p e p (x ; y ) = 0 ; w h e re p (x ; y ) = a x 2 + by 2 + c x y + d x + e y + f : O b se rv e th a t p h a s 3 + 2 + 1 = 6 p a ra m e te rs. If w e m u ltip ly p b y a n o n -ze ro co n sta n t, th e c u rv e re m a in s Figure 4. (a) (b) RESONANCE October 2008 919

Five points in general position are needed to fix a second degree curve. th e sa m e, so F is a v e -p a ra m e te r fa m ily. T h e g eo - m e trica l im p lica tio n o f th is is th a t v e p o in ts in ge n era l po sitio n a re n eed ed to x a seco n d d egree cu rve. (In c o n - tra st, th e fa m ily o f c ircles h a s th ree d eg re es o f fre ed o m, a n d th re e p o in ts in g e n e ra l p o sitio n a re n e ed ed to x a circ le. In b o th c a ses, `g en e ra l p o sitio n ' m e a n s th a t n o th re e p o in ts lie in a stra ig h t lin e.) If w e x fo u r p o in ts a n d a llo w th e fth o n e to v a ry, w e g et a o n e p a ra m ete r fa m ily (a `p e n c il o f c o n ic s'), a s th e sk etch in F ig u re 4 b sh o w s. A sim p le b u t e x trem e ly u se fu l co ro lla ry to th e a b o v e o b - se rv a tio n is th e fo llo w in g : If C 1 a n d C 2 a re tw o seco n d d e- gree cu rves pa ssin g th ro u gh fo u r given po in ts A ; B ; C ; D, w ith equ a tio n s p 1 (x ; y ) = 0 a n d p 2 (x ; y ) = 0, re spectively, th en th e equ a tio n o f a n y o th er seco n d d egree cu rve C 3 pa ssin g th ro u gh A ; B ; C ; D m a y be w ritten in th e fo rm 1 p 1 (x ; y ) + 2 p 2 (x ; y ) = 0, w h ere 1 ; 2 a re rea l co n - sta n ts, n o t bo th zero. S im ila r sta te m en ts m a y b e m a d e a b o u t th e p e n cil o f lin es p a ssin g th ro u g h a x e d p o in t, o r a b o u t th e fa m ily o f c irc le s p a ssin g th ro u g h tw o g iv e n p o in ts. (T h is p rin cip le e x te n d s to cu b ic s a s w e ll, b u t w e sh a ll stu d y th is o n ly in P a rt 2.) 3. P a s c a l's H e x a g r a m T h e o r e m W e n o w p ro v e a fa m o u s a n d im p o rta n t th eo rem a b o u t th e co n ic sec tio n s, d u e to B la ise P a sc a l (1 6 2 3 { 1 6 6 2 ). T h e o r e m 2 (P a sc a l). T h e o p po site sid es o f a h e xa g o n in scribed in a co n ic in tersect in th ree co llin ea r po in ts. (S e e F ig u re 5.) T h a t is, if six d istin c t p o in ts A ; B ; C ; D ; E ; F lie o n a co n ic C, th e n th e p o in ts o f in te rse ctio n P = Ã A! B \ Ã D! E, Q = Ã B! C \ Ã E! F, R = Ã C! D \ Ã F! A lie in a stra ig h t lin e. It is n o t k n o w n h o w P a sc a l p ro v e d th e th eo re m, o r h o w h e h it u p o n it (w h ich h e d id a t th e a g e o f six te en ). A b o o k h e w a s w ritin g o n th e co n ic sec tio n s circu la te d fo r so m e y ea rs a m o n g th e p ro m in e n t m a th e m a tic ia n s o f E u - ro p e, in d ra ft co p y, a n d th en w a s lo st to m a n k in d. 920 RESONANCE October 2008

P a sc a l's th eo rem is a th eo rem o f p ro je ctiv e g e o m e try : it a llo w s th e p o in ts P ; Q ; R to lie o n th e `lin e a t in n ity '. T h u s, if à A! B k à D! E, th en P is `a p o in t a t in n ity '. In th is ca se th e th e o re m im p lie s th a t à Q! R k à A! B. If it h a p p e n s th a t à A! B k à D! E a s w e ll a s à B! C k à E! F, th en th e th eo rem a sserts th a t à C! D k à F! A. (N o w a ll th re e o f P ; Q ; R lie o n th e lin e a t in n ity.) A n im p o rta n t co n se q u e n c e o f th e p ro jec tiv e v ie w p o in t is th a t th e lin e a t in n ity d o e s n o t h a v e a n y sp e cia l sta tu s; it is trea te d o n p a r w ith e v e ry o th e r lin e. S o in th e p ro je ctiv e p ro o f, it is o f n o c o n se q u en ce if so m e p a irs o f lin e s a re p a ra lle l to e a ch o th er; th e w o rd in g o f th e p ro o f re m a in s e x a c tly th e sa m e. T o im p lem e n t th is a p p ro a ch, w e u se p ro jective coo rd in a tes, in w h ich p o in ts a re d e n o te d u sin g trip les [x ; y ; z ] o f re a l n u m b e rs. H ere a re th e b a sic ru le s g o v e rn in g th e se trip le s: (i) x ; y ; z a re n o t a ll z e ro ; (ii) [k x ; k y ; k z ] d e n o te s th e sa m e p o in t a s [x ; y ; z ] fo r a n y rea l n u m b er k 6= 0. T h e u n d e rsta n d in g is th a t if z 6= 0, th e n [x ; y ; z ] co rresp o n d s to th e p o in t w ith ca rtesia n c o o rd in a te s (x = z ; y = z ), a n d if z = 0 th e n [x ; y ; z ] lie s o n th e lin e a t in n ity. T h e lin e w ith c a rte sia n eq u a tio n x + y = 1 a cq u ires th e e q u a tio n x + y z = 0 in th is sy ste m, w h ile th e c ircle w ith c a rte sia n e q u a tio n Figure 5. An important consequence of the projective viewpoint is that the line at infinity does not have any special status; it is treated on par with every other line. RESONANCE October 2008 921

x 2 + y 2 x y = 1 a c q u ire s th e e q u a tio n x 2 + y 2 x z y z z 2 = 0. N o te th a t th e se e q u a tio n s a re h o m og en eo u s. T h e lin e a t in n ity h a s th e eq u a tio n z = 0. P roo f o f P a sca l's T h eo rem. L e t L A B (x ; y ; z ) = 0, L B C (x ; y ; z ) = 0, L C D (x ; y ; z ) = 0, : : : b e th e e q u a tio n s o f à A! B, à B! C, à C! D : : :, re sp e c - tiv ely, w h e re L A B, L B C, L C D, : : : a re lin e a r e x p re ssio n s in x ; y ; z. L e t f (x ; y ; z ) = 0 b e th e e q u a tio n o f th e c o n ic, w h e re f is a p o ly n o m ia l in x ; y ; z o f d e g re e 2. T h e v a rio u s L 's a n d f a re h o m o g en e o u s e x p ressio n s. S in ce C p a sses th ro u g h A ; B ; C ; D, a n d so d o th e tw o p a ir-o f-stra ig h t-lin e s c o n ic s à A! B [ à C! D a n d à A! D [ à B! C (see F igu re 6 ), th e re ex ist c o n sta n ts ; 0 su ch th a t f = L A B L C D + 0 L A D L B C : (1 ) S im ila rly, sin ce C p a sse s th ro u g h A ; E ; F ; D, a n d so d o th e tw o c o n ics à A! D [ à E! F a n d à D! E [ à A! F, th e re e x ist c o n - sta n ts ; 0 su ch th a t f = L A D L E F + 0L D E L A F : (2 ) F ro m (1 ) a n d (2 ) w e g e t L A B L C D + 0 L A D L B C = L A D L E F + 0L D E L A F ; ) L A D ( 0 L B C L E F ) = 0L D E L A F L A B L C D : (3 ) Figure 6. 922 RESONANCE October 2008

L e t p o ly n o m ia ls g ; h b e d e n e d a s fo llo w s: ½ g = 0L D E L A F L A B L C D ; h = 0 L B C L E F : (4 ) S in ce it c a n n o t h a p p e n th a t h is id e n tica lly 0 (th is w o u ld m a k e à B! C c o in c id e w ith à A! D, w h ich is n o t a llo w e d b y th e h y p o th e se s o f th e th e o re m ), it m u st b e th a t h h a s d eg ree 1. T h e n th e e q u a tio n h = 0 rep re se n ts a stra ig h t lin e. N o w o b serv e th a t: ² P 2 à A! B a n d P 2 à D! E, so g (P ) = 0. S in c e P 62 à A! D, it fo llo w s th a t h (P ) = 0. ² R 2 à C! D a n d R 2 à A! F, so g (R ) = 0. S in ce R 62 à A! D, it fo llo w s th a t h (R ) = 0. ² Q 2 à B! C a n d Q 2 à E! F, so h (Q ) = 0. When C is a pair! of straight lines we get the result known as Pappus s theorem. S o P ; Q ; R lie o n th e lin e 0 L B C L E F = 0, a n d a re th u s c o llin e a r. R e m a r k s ² T h e p ro o f d o e s n o t c o n sid e r sep a ra te ly th e ca se s o f p a ra lle lism. (It is n o t req u ired, a s p er th e re m a rk s m a d e a b o v e.) ² P a sca l's th e o re m is e x trem e ly w id e in its sc o p e : th e h e x a g o n A B C D E F m a y b e n o n -c o n v e x a n d / o r selfin terse c tin g, a n d th e c o n ic C itse lf m a y b e a n y se c - o n d d eg ree lo cu s. W h e n C is a p a ir o f stra ig h t lin e s w e g et th e re su lt k n o w n a s P a p p u s's th e o re m. ² W e m a y ev en a llo w so m e p a irs o f p o in ts to co in c id e; in th is ca se w e g et `lim itin g c a se s' o f th e th eo rem b y re p la cin g `lin e s' b y `ta n g e n ts' a s n e e d e d. F o r e x a m p le if w e le t A co in c id e w ith B, w e g e t th e fo l- lo w in g sta tem e n t, in w h ich t X d en o tes th e ta n g en t to th e co n ic a t a n y p o in t X o n it: If B ; C ; D ; E ; F a re ve d istin ct po in ts o n a co n ic C, th en th e po in ts P = t B \ à D! E, Q = à B! C \ à E! F, R = à C! D \ à E! F lie in a stra igh t lin e. RESONANCE October 2008 923

If ABCDEF is a hexagon formed by six lines that are all tangent to a conic, then the lines AD, BE, CF concur. ² A sim ila r p ro o f m a y b e d ev ised fo r B ria n ch o n 's th e - o rem : If A B C D E F is a h exago n fo rm ed by six lin es th a t a re a ll ta n g en t to a co n ic, th e n th e lin e s à A! D, Ã! B E, à C! F co n c u r. 4. T h e B u tt e r y T h e o r e m T h e b u tte r y th e o re m rst a p p e a re d a s a p ro b lem in th e e a rly 1 8 0 0 's, a n d o n e o f th e ea rly p ro o fs is d u e to W G H o rn er w h o is b e tter k n o w n fo r a n a lg o rith m in p o ly n o m ia l a rith m e tic. T h e o r e m 3 (B u tter y T h eo rem ). L et P Q be a ch o rd o f circle K, a n d let M be its m id po in t. T h ro u gh M tw o o th er ch o rd s A B a n d C D a re d ra w n. L et A D a n d B C cu t Ã! P Q in E a n d F, respectively. T h en M is th e m id po in t o f E F. (S e e F igu re 7.) A la rg e n u m b e r o f e le g a n t p ro o fs o f th e B u tter y T h eo - rem h a v e a p p e a re d o v e r th e y e a rs, in clu d in g m a n y `p u re g e o m e try ' p ro o fs; b u t th e fo llo w in g p ro o f is p a rtic u la rly ch a rm in g, fo r it sim u lta n eo u sly ca sts th e th eo rem in a m o re g en e ra l se ttin g a n d m a k e s it e a sy to fo rm u la te sp e - cia l c a se s o f in te re st. It is b a se d y et a g a in o n th e a sse r- tio n s m a d e in S e c tio n 2. Figure 7. 924 RESONANCE October 2008

T h e o r e m 4 (B u tter y T h eo rem fo r C o n ic s). L et P Q be a ch o rd o f a co n ic K, a n d let M be its m id po in t. T h ro u gh M let tw o o th er ch o rd s A B a n d C D be d ra w n. L et A D a n d B C cu t à P! Q in E a n d F, respectively. T h en M is th e m id po in t o f E F. (S ee F igu re 8.) P roo f. W e u se a c a rte sia n se ttin g, w ith Ã! P Q a s th e x -a x is, a n d M a s th e o rig in. L e t K h a v e eq u a tio n p (x ; y ) = 0, w h e re p (x ; y ) = a x 2 + b y 2 + c x y + d x + e y + f. T h e in te rse ctio n s o f K w ith th e x -a x is a re fo u n d b y so lv in g th e eq u a tio n s p (x ; y ) = 0, y = 0, i.e., a x 2 + d x + f = 0 : (5 ) S in ce M is th e m id p o in t o f P Q, th e su m o f th e ro o ts o f (5 ) is 0, w h ich im p lie s th a t d = 0. H e n ce, th e coe ± cien t o f th e x -term in p (x ; y ) is zero. T h is a ssertio n a lso h o ld s fo r th e lin e -p a ir co n ic K 0 = Ã! A B [ à C! D, b ec a u se th e lin e s à A! B a n d à C! D p a ss th ro u g h th e o rig in. T h a t is, if th e equ a tio n o f K 0 is q (x ; y ) = 0, th en th e coe± cien t o f th e x -term in q (x ; y ) is zero. S in ce K a n d K 0 sh a re th e fo u r p o in ts A ; B ; C ; D, th e sa m e sta tem en t is tru e fo r a n y co n ic th a t pa sses th ro u gh A ; B ; C ; D. T h is is b ec a u se th e e q u a tio n o f a n y su ch co n ic is o f th e fo rm r p (x ; y ) + s q (x ; y ) = 0, w h e re r ; s a re re a l n u m b ers. Figure 8. RESONANCE October 2008 925

A surprising consequence of Pascal s theorem is that it allows us to define a group on the points of any nondegenerate conic. GENERAL ARTICLE In p a rticu la r it is tru e fo r th e lin e-p a ir c o n ic K 00 = à A! C [ Ã! B D. H en c e, th e su m o f th e ro o ts o f th e in te rse c tio n s o f K 00 w ith th e x -a x is is 0. T h a t is, M is th e m id p o in t o f E F. H e re a re tw o ty p ic a l `sp ec ia l c a se s' w h o se p ro o fs w e le a v e to th e re a d er: T h e o r e m 5. L et A B C D be a cy clic qu a d rila tera l w h o se circu m circle K h a s A C a s a d ia m eter, a n d O a s its cen - ter. L et th e ta n gen t to K a t A m eet à B! D a t P ; let à P! O m eet à C! B in E, a n d à C! D in F, respectively. T h en O is th e m id po in t o f E F. (S ee F igu re 9.) T h e o r e m 6. L et A B, C D be segm en ts in tersectin g a t a po in t P, a n d let Q a n d R be po in ts o n A B a n d C D, respectively, su ch th a t A P = Q B, a n d C P = R D. L et Ã! Q R m eet A D in U, a n d B C in V. T h en U R = Q V. (S ee F igu re 1 0.) 5. G r o u p o n a C o n ic Figure 9 (left). Figure 10 (right). In c lo sin g w e p o in t o u t a ra th e r su rp risin g c o n se q u en ce o f P a sc a l's th e o re m : it a llo w s u s to d e n e a g ro u p o n th e p o in ts o f a n y n o n -d e g e n e ra te c o n ic. L et N b e a n y x e d p o in t o f su ch a co n ic K ; th is w ill se rv e a s th e n e u tra l p o in t, i.e., th e id e n tity e lem e n t o f th e g ro u p. W e d e n e 926 RESONANCE October 2008

(a) (b) th e b in a ry o p era tio n th u s: if P ; Q a re p o in ts o n K, w e d ra w th ro u g h N a lin e p a ra lle l to à P! Q ; th en P Q is th e p o in t w h ere th is lin e in te rse cts K a g a in. T h e co n stru c tio n is a s d e p icte d in F igu re 1 1 a. Figure 11. It is ea sy to ch e ck th e fo llo w in g a ssertio n s: ² T h e o p era tio n is w ell d e n e d. F o r, th e lin e th ro u g h N p a ra llel to à P! Q w ill in te rse ct K a se co n d tim e, sin c e K is a sec o n d d e g re e cu rv e. ² P Q = Q P fo r a ll p a irs o f p o in ts P ; Q 2 K ; so is c o m m u ta tiv e. ² If P = Q, th e n Ã! P Q is ta n g en t to K a t P. S o to n d P P, w e d ra w a lin e th ro u g h N p a ra lle l to th e ta n g e n t t P to K a t P ; th e n its se c o n d p o in t o f in terse c tio n w ith K is P P. S ee F igu re 1 1 b. ² W e h a v e N N = N, a n d P N = P fo r a ll P 2 K. ² If th e lin e th ro u g h N p a ra lle l to Ã! P Q is ta n g e n t to K a t N, th e n R = N, so w e w rite P Q = N. T h is e n a b le s u s to n d in v erses. If w e in v o k e P a sca l's th eo rem in th is co n g u ra tio n w e n d fa irly e a sily th a t P (Q R ) = (P Q ) R fo r a n y th re e p o in ts P ; Q ; R o n K. T h is im p lie s th e fo llo w in g resu lt. T h e o r e m 7. T h e pa ir (K ; ) is a n a belia n gro u p, w ith N a s th e id en tity elem en t. RESONANCE October 2008 927

It is in te restin g to c la ssify th e g ro u p s co rresp o n d in g to d i e ren t c o n ic s. H e re a re th e m a in re su lts, w h o se p ro o fs w e le a v e to th e rea d e r: 1. If K is eith e r a n ellip se o r a c ircle, th e n (K ; ) is iso m o rp h ic to th e m u ltip lic a tiv e g ro u p o f c o m p le x n u m b e rs w ith u n it m a g n itu d e, i.e., iso m o rp h ic to th e g ro u p R = 2 ¼ Z. 2. If K is a p a ra b o la, th e n (K ; ) is iso m o rp h ic to th e a d d itiv e g ro u p o f re a l n u m b e rs, (R ; + ). 3. If K is a h y p e rb o la, th e n (K ; ) is iso m o rp h ic to th e m u ltip lic a tiv e g ro u p o f n o n -z ero re a l n u m b e rs, (R? ; ). T h e re a re n u m e ro u s q u estio n s o f in te re st w h ich c a n b e ex p lo re d in th is re g a rd, in c lu d in g so m e w h ich a re o f a n u m b e r-th e o re tic n a tu re. T h e se a rise w h en o n e re g a rd s th e co n ic a s d e n e d o v e r so m e n ite eld ra th er th a n th e e ld o f re a l n u m b ers; e.g., o v e r th e in te g e rs m o d u lo p fo r so m e p rim e p. T h e c o n ic in su ch a ca se is n o t a v isu a liz a b le o b jec t, a n d o n e h a s to stu d y it a lg e b ra ic a lly ra th e r th a n g eo m e trica lly. H e re is a ty p ic a l resu lt: If K is th e co n ic x 2 2 y 2 = 1 o ve r th e eld o f in tege rs m od u lo a p rim e p, th en (K ; ) is a cy clic gro u p ; its o rd er is p 1 if 2 is a qu a d ra tic resid u e m od u lo p, a n d p + 1 if 2 is a qu a d ra tic n o n re sid u e m o d u lo p. H o w ev er w e sh a ll n o t d w ell o n th is to p ic h ere. T h e re a d e r is re ferre d to [4 ] fo r fu rth e r d e ta ils. Suggested Reading Address for Correspondence Shailesh A Shirali Rishi Valley School Rishi Valley 517 352 Madanapalle, AP, India. Email: shailesh.shirali@gmail.com [1] H S M Coxeter, Introduction to Geometry, 2nd edition, Wiley, 1989. [2] H S M Coxeter and S L Greitzer, Geometry Revisited, Math. Assoc. Amer., 1st edition, 1967. [3] S L Loney, The Elements of Coordinate Geometry, in two volumes, MacMillan India, 2000. [4] Shailesh Shirali, Groups Associated With Conics, The Mathematical Gazette, (to appear February 2009). [5] Eric W Weisstein, Conic Section, From MathWorld A Wolfram Web Resource, http://mathworld.wolfram.com/conicsection.html 928 RESONANCE October 2008