1156 RESONANCE December 2008GENERAL ARTICLEKeywordsFelix Klein, Erlanger program,isometry, group, invariant, halfturn, Napol eon' s theorem,Morley's theorem, affine map-ping.Algebraic Methods in Plane Geometry3. The Use of MappingsShailesh A ShiraliShailesh Shirali heads aCommunity MathematicsCenter at Rishi ValleySchool (KFI). He has adeep interest in teachingand writing aboutmathematics at the highschool/post school levels,with particular emphasison problem solving and thehistorical aspects of thesubject.Part 1. The Use of Conic Sec-ti ons, Resonance, Vol .13,No.10, pp.916928, 2008.Part 2. Cubic Curves,Reso-nance, Vol.13, No.11, pp.10271040.In th is a rtic le w e e x a m in e th e ro le o f m a p p in g s ine le m e n ta ry g e o m e try . A fte r m a k in g so m e c o m -m e n ts a b o u t th e E rla n g e n p ro g ra m m e in itia te db y F e lix K le in in 1 8 7 2 , w h e re in h e p ro p o se d aw a y o f stu d y in g g e o m e trie s b a se d o n th e u n d e rly -in g tra n sfo rm a tio n g ro u p s, w e se e h o w th e o re m slik e V o n A u b e l's th e o re m a n d N a p o le o n 's th e o -re m c a n b e p ro v e d in a n e le g a n t m a n n e r u sin gsim ila rity m a p p in g s, a n d h o w so m e c o n stru c tio np ro b le m s m a y b e so lv e d u sin g iso m e trie s. A t th ee n d w e p re se n t a re c e n t p ro o f b y A la in C o n n e so f \ M o rle y 's M ira c le " , b a se d o n a n e tra n sfo r-m a tio n s.1 . F e lix K le in 's E rla n g e r P ro g ra m m eIn 1 87 2, w h en th e g rea t G erm a n m a th em aticia n F elixK lein w as a p p o in ted p rofessor at E rla n gen , h e p rep aredan in au g u ral ad d ress in w h ich h e set fo rth a u n i ed v iewof a ll of geom etry. T h is v iew over tim e b ecam e k n ow n a sth e E rlan ger program , an d it h a d a p rofo u n d in u en ceon th e d ev elop m en t of geom etry. K lein 's th esis w a s th atgeom etry is the stu dy of those properties of a space thatrem ain in varian t un der som e given grou p of tran sform a-tion s.T o illu stra te w h at th is m ean s, con sid er th e grou p ofisom etries in th e p lan e. A m ap p in g f : R2! R2is saidto b e an isom etry if it p reserv es d ista n ces: d (f (P );f (Q ))= d (P ;Q ) fo r a ll p a irs o f p oin ts P ;Q 2 R2, w h ere d is th ed istan ce fu n ction . R e ectio n in a lin e is an ex am p le ofsu ch a m a p p in g. It is clear th a t su ch a m ap p in g m u st b eon e to o n e, an d th erefo re in vertib le. In th e n ex t sectio nw e sh all ex am in e su ch m ap p in gs in m o re d eta il, b u t forn ow w e n ote o n ly th at th e set Iso o f all su ch m ap p in g s1157 RESONANCE December 2008GENERAL ARTICLEform s a g rou p in a n a tu ral m an n er, u n d er fu n ctio n alco m p osition ; fo r, if f ;g ;h 2 Iso , th enf g 2 Iso ; th ein verse f 1of f is in Iso ; th e id en tity m ap Id is in Iso ;an d f (g h ) = (f g ) h , th is rela tio n b ein g tru efor an y th ree fu n ction s fo r w h ich all th e com p o sitio n sare w ell d e n ed . A s all th e grou p ax io m s are sa tis ed ,(Iso ; ) is a grou p . It is ca lled th e E uclidean group.If w e allow elem en ts of th is gro u p to act o n an arb itrarygeom etric o b ject in R2, w h at fea tu res w ill th e im a gessh a re w ith on e an o th er? B y d e n ition , d istan ces stayth e sam e. T h is im p lies th at n o tio n s lik e collin earity stayin tact: if p oin ts A ;B ;C lie in a stra ig h t lin e, so d o th eirim ag es A 0;B 0;C 0. L ik ew ise for perpen dicularity : if l;mare lin es p erp en d icu la r to each o th er (l ? m ), th en th eirim ag es l0;m 0m ain ta in th at relation (l0? m 0). M ore g en -era lly, an gles stay u n ch a n ged : \ (l;m ) = \ (l0;m 0); to seew h y, w e on ly n eed to u se th e cosin e ru le. S u ch p ro p er-ties are in varian t w ith resp ect to th e E u clid ean g rou p ,an d u n d er K lein 's E rla n ger p ro gram m e, th e stu d y ofall su ch p rop erties m ay b e said to con stitu te E u clid ea ngeom etry.T h e n o tio n o f w h a t `co n stitu tes E u clid ean geom etry 'm ay seem im p recise, so w e give an ex am p le o f a p rop ertyth at d o es n ot b elon g h ere. Im agin e th at on e in tro d u cesa n o tio n of balan ced : `A trian gle is b a la n ced if its b ase isp ara llel to th e x -ax is'. B y u sin g a rotation (w h ich is a nisom etry ), w e so o n see th a t th ere a re con g ru en t p a irs oftria n gle in w h ich on e trian g le is b a la n ced an d th e oth eris n ot. H en ce, su ch a p ro p erty is n o t in varian t w ith re-sp ect to th e E u clid ea n g rou p , an d so d o es n ot b elo n g toE u clid ean g eo m etry.If w e en large th e E u clid ean gro u p b y in clu d in g all en -largem en ts, w e g et th e gro u p of sim ila rity tran sfo rm a -tio n s, or th e group of m appin gs in the plan e w hich pre-serve collin earityan dleave an gles u n altered. In th ela n gu a ge o f com p lex n u m b ers, su ch m ap s com e in tw oform s, z 7! a z + b a n d z 7! a z + b (a ;b 2 C, a 6= 0;Geometry is thestudy of thoseproperties of aspace that remaininvariant under agiven group oftransformations.Kleins Erlangerprogram had aprofound influenceon thedevelopment ofgeometry.1158 RESONANCE December 2008GENERAL ARTICLEin th e ca se of iso m etries, w e h ave th e fu rth er con d itio nja j = 1). T h e stu d y o f p rop erties left in varian t b y th isgro u p con stitu tes sim ilarity g eo m etry. T h e variou s th e-orem s of sch o ol g eo m etry b elon g h ere.T h e grou p o f sim ilarities is a su b g rou p o f th e group ofa n e tran sform ation s, u n d er w h ich th e p ro p erties ofco llin ea rity an d p a rallelism a re p reserv ed , b u t n ot d is-tan ces an d an g les. (T o see h ow th is m ay com e ab ou t,th in k o f a shear.) A n d th e sim ila rity gro u p in tu rn isa su b g rou p o f th e grou p of projective tran sform ation s,u n d er w h ich th e p ro p erties of collin earity an d in cid en ceare p reserved , b u t n o t p a rallelism . It m ay seem th at a llis lo st b y th is stag e, b u t th is is n ot so: cross ratios arep reserv ed , a n d th is g ives rise to m a n y b eau tifu lth eorem sof p ro jectiv e g eo m etry ; e.g ., th e th eorem s of P asca l a n dB rian ch on .2 . C la sse s o f Iso m e trie sO b v iou s ch oices for isom etries are th e follow in g : re ec-tio n in a n y lin e; rota tion ab ou t an y p o in t, b y an y a n gle;d isp la cem en t th ro u gh an y vector. It m ay b e sh ow n th atco m p osition s of th ese m ap s y ield all p o ssib le iso m etries.In fact, th e fo llow in g stron g er resu lt is tru e: A n y isom -etry is a com position of n o m ore than three re ection s.T h is is th e `th ree re ection s th eorem '.T h e p ro of o f th is is b a sed on th e ob servation th at a nisom etry is k n ow n if w e k n ow th e im ages of an y th reen on -co llin ea r p o in ts. T h is in tu rn fo llow s from elem en -tary con g ru en ce th eorem s. F o r, let f b e an isom etryw h ich ta kes A to A 0an d B to B 0. L et C b e an y p o in tn ot on !A B . If its im age u n d er f is C 0, th en 4 A B C =4 A 0B 0C 0. H en ce, C 0lies o n th e circle w ith cen ter A 0a n drad iu s jA C j, an d a lso on th e circle w ith cen ter B 0a n drad iu s jB C j. T h is m ea n s th a t th ere a re ju st tw o p o ssib lelo cation s fo r C 0. O n ce th is lo catio n is x ed , th e lo catio nof th e im a ge o f a n y fou rth p o in t g ets x ed b eca u se w ek n ow its d istan ce from each of A 0;B 0;C 0; n o m ore ch oiceis left.Any isometry is acomposition of nomore than threereflections.1159 RESONANCE December 2008GENERAL ARTICLESymbol DescriptionD vDisplacement t hrough a vect or vR A ;Rot at ion about a point A, t hrough an angle H AHalf t urn about a point A ( hence, H A= R A ;1 8 0 )M`Reect ion in a line `E (A ;k )Enlargement about a point A, by a scale fact or k (k 6= 0)Table 1. Some classes ofmappings.Figure 1. (left) Reflectionsin two parallel mirrors.Figure2. (right) Reflectionsintwointersectingmirrors.Isom etries a re said to b e direct if th ey p reserv e orien -tation (i.e, th e sen se of clo ck w ise-an ticlo ck w ise); ex am -p les: rota tio n s a n d d isp lacem en ts. If n ot, th ey are in di-rect; ex am p le: re ection s. T h e th ree re ection s th eo remim p lies th e fo llow in g: E very direct isom etry is either adisplacem en t or a rotation ; every in direct isom etry iseither a re ection , or a re ection com posed w ith a dis-placem en t, or a re ection com posed w ith a rotation .In th e lan gu age o f com p lex n u m b ers: isom etries o f th eform z 7! a z + b (ja j = 1 ) a re d irect, an d th o se o f th eform z 7! a z + b (ja j = 1) are in d irect. F or b o th fo rm s,th e ro tation al co m p on en t of th e isom etry is ca p tu red b yth e arg u m en t of a .T h e d irect isom etries form a su b g rou p o f in d ex 2 inth e g rou p (Iso ; ). W ith th e n ota tion a s in T able 1 , w e n d vario u s relation sh ip s (T h eorem s 1 {3) am on g th eseclasses of m a p p in g s.T h e o re m 1 (C om p o sition of re ectio n s). L et l;m bedistin ct lin es. T hen Mm Ml is a displacem en t if l k m ,an d a rotation if l 6k m . (Mm Ml m ea n s: \ R e ect in l,an d th en in m ".) In the form er case, the displacem en tis through tw ice the directed distan ce from l tom ; inthe latter case the an gle of rotation is tw ice the directedan gle \ (l;m ). (S ee F igu res 1 an d 2.)1160 RESONANCE December 2008GENERAL ARTICLEC o ro lla ry 1 .1 . T w ore ection s com m ute w ith eachother if an d on ly if their axes are perpen dicu lar to eachother, or they coin cide.T h e o re m 2 (C om p osition of rotation s). T he com posi-tion of tw o rotation s is either a rotation or a displace-m en t: if A ;Bare arbitrary poin ts, an d ; are arbitraryan gles, then R A ; R B ; is a rotation by + about som epoin t C , un less + is a m ultiple of 360, in w hich casethe com posite m ap is a displacem en t.C o ro lla ry 2 .1 . T he com position of an odd n um ber ofhalf turn s is a half turn .T he com position of an evenn um ber of half turn s is a displacem en t.T h e o re m 3 (C h ara cteriza tio n o f sim ilarity m ap s).Am ap that preserves the relation ship of equality of len gthsis a sim ilarity m ap. T hat is, if f : R2! R2is su ch thatfor every collection of fou r poin ts P ;Q ;R ;S w e have thefollow in g relation ,d (P ;Q ) = d (R ;S ) = ) d (f (P );f (Q )) = d (f (R );f (S )) ;thenf is a sim ilarity m ap. (T h is m ean s th a t fo r an yth ree p o in ts P ;Q ;R , th e tria n gle w ith vertices f (P );f (Q );f (R ) is sim ilar to 4 P Q R .)T h eorem 3 is far from ob v io u s. W e give o n ly th e p ro ofof T h eo rem 2 .G iven tw o d istin ct p oin ts A ;B a n d tw o an g les, ; , w em u st n d w h at th e co m p ositio n R B ; R A ;d o es. L et lb e th e lin e !A B , let m b e th e lin e th ro u gh A su ch th at\ (m ;l) =12 , an d let n b e th e lin e th rou g h B su chth at \ (l;n ) =12 (th ese a re d irected a n gles; see F igure3); th en \ (m ;n ) =12 ( + ). T h eorem 1 tells u s th atR A ; = Ml Mman d R B ; = Mn Ml. H en ceR B ; R A ; = Mn Ml Ml Mm= Mn Mm :If + 0 (m o d 36 0) (th is is th e ca se, for ex am p le,if = ), th enmk n , an d th e co m p osite m ap is aThe composition ofan odd number ofhalf turns is a halfturn. Thecomposition of aneven number ofhalf turns is adisplacement.1161 RESONANCE December 2008GENERAL ARTICLEFigure 3. Composition oftwo rotations.displacem en t. If n ot, th en m an d n m eet a t so m e p o in tC , so th a t Mn Mm= R C ; + . In th is ca se, the com positem ap is equivalen t to the rotation R C ; + .O r, u sin g com p lex n u m b ers: let f (z ) = a z + b a n dg (z ) = cz + d rep resen t tw o rota tio n s, w ith ja j = 1 = jcj,a 6= 1, c 6= 1 ; th enf g (z ) = a cz + a d + b. T h e factth at ja cj = 1, an d th e p resen ce o f z rath er th an z , tellsu s th at th is is a d irect isom etry. If arg a + arg c 0(m o d 2 ) th en f g is a d isp lacem en t; else, it is a ro -tation . (R em ark. T h is p ro o f is m ore com p act th a n th eea rlier on e, b u t th a t p ro o f also y ield s th e lo ca tio n of th ecen ter of th e co m p osite m ap , an d w e sh a ll see sh o rtlyh ow th is k n ow led ge can b e of u se.)3 . G ro u p s A sso c ia te d w ith Iso m e trie sA s n oted ea rlier, th e set Iso of all iso m etries in th e p lan eform s th e E u clid ean gro u p . T h is h a s va rio u s su b grou p sof in terest. C on sid er th e isom etries th a t x a p a rticu larp o in t O . T h ese a re a lso th e sy m m etries of an y x edcircle w ith cen ter O . T h ese iso m etries form a su b gro u pof Iso ; it is called th e orthogon al group an d is d en oted b yO (2). T h e co m p on en ts o f th is gro u p are th e rota tio n sR O ;, w ith tak in g all p o ssib le valu es, a n d th e re ectio n sMl, for allp ossib le lin es l th ro u gh O . T h e rota tion s alon eco n stitu te a n orm al su b g rou p of O (2 ), ca lled th e specialorthogon al group a n d d en o ted b y S O (2); it is o f in d ex 2in O (2 ). It is n ot h a rd to sh ow th a t O (2 ) is isom o rp h icto th e m u ltip licative g rou p of a ll 2 2 m a trices A forw h ich A A T= I2, w h ere I2 is th e 2 2 id en tity m atrix ,w h ile S O (2) is isom orp h ic to th e m u ltip lica tiv e gro u p ofall 2 2 m atrices A for w h ich A A T= I2, d et A = 1 . T h eelem en ts o f S O (2) a ll h av e th e formThe compositionof two rotations iseither a rotation ora displacement.1162 RESONANCE December 2008GENERAL ARTICLEcos sin sin cos ( 2 R);w h ile th e elem en ts of O (2) h ave th e form scos sin sin cos ;co s sin sin cos ( 2 R):T h e q u otien t g rou p O (2)= S O (2 ) is isom orp h ic to(f 1; 1g ; ).A n o th er im p orta n t su b gro u p of Iso is th e g ro u p D ofall d isp lacem en ts in th e p lan e; th is is essen tia lly th e a d -d itive grou p of all vecto rs in th e p la n e. In fa ct, D isa n orm al su b gro u p of Iso . T o sh ow th is w e m u st sh owth at if f is a d isp lacem en t, an dg is an iso m etry, th eng1 f g is a d isp lacem en t. U sin g com p lex n u m b ersth is b ecom es ea sy : let f (z ) = z + a an d g (z ) = bz + c,w h ere a ;b;c 2 C, jbj = 1 ; th en :g1 f g (z ) = g1 f (bz + c)= g1(bz + c + a ) =bz + c + a cb= z +ab ;w h ich is a d isp lacem en t. (N o te th at th e p ro o f sh ow sth at D is a n orm al su b g ro u p of th e sim ila rity g rou p .)T h e q u otien t g rou p Iso = D is iso m o rp h ic to O (2).4 . T h e U se o f M a p p in g s in C o n stru c tio n sT h e follow in g p rob lem y ield s n icely to th e algeb ra ofm a p p in gs: G iventhe m idpoin ts of the sides of an n -sided polygon , inproper order, con stru ct the polygon .In th e case of a trian g le (n = 3 ) th e p rob lem is ea silysolved u sin g th e m id p o in t th eorem , b u t w e esch ew th isap p ro ach an d lo o k a t th e p rob lem algeb raically, a s th isen ab les u s to n d a u n i ed ap p ro ach th at w o rk s for a lln 3 .S u p p ose th en th a t n = 3; let th e g iv en p o in ts b e P ;Q ;R .W e m u st n d p oin ts A ;B ;C su ch th at P ;Q ;R are th em id p oin ts o f B C , C A , A B , resp ectiv ely. C o n sid er th eThe group of alldisplacements is anormal subgroupof the group ofisometries.1163 RESONANCE December 2008GENERAL ARTICLEFigure 4.ConstructingABC from the midpointsP, Q, R of its sides.follow in g m a p : f = H R H Q H P . A s f is a com -p o sition o f an o d d n u m b er of h alf tu rn s, it is itself ah alf tu rn . N ow ob serve th e e ect of f on B : th e m a p sH P ;H Q ;H Rap p lied in tu rn take B th rou g h th e orb itB 7! C 7! A 7! B , im p ly in g th at f (B ) = B . H en ce, B isth e cen ter of f . T o lo cate B , it su ces to ap p ly f to an y`test p oin t' X : if f m ap s X to Y , th en B is th e m id p o in tof X Y . H av in g fou n d B , w e n ow n d C an d A u sin g th erelation s H P (B ) = C , H Q (C ) = A . F igure 4 illu stratesth e solu tion .N o te th a t: (a) a so lu tion to th e ab ov e p ro b lem ex istsfor an y th ree given p oin ts P ;Q ;R (if th ey are co llin ear,th en A ;B ;C to o w ill b e co llin ear); an d : (b ) th e sa m eap p ro ach w ork s for a n y o d d va lu e o f n (n 3 ). F or ex -am p le, su p p ose w e are given ve p o in ts P ;Q ;R ;S ;T , a n dw e w an t v e p o in ts A ;B ;C ;D ;E su ch th a t P ;Q ;R ;S ;Tare, resp ectively, th e m id p oin ts of B C , C D , D E , E A ,A B . L et f b e th e h alf tu rn H T H S H R H Q H P . T h ecen ter of f is B , a n d it is fou n d in th e sa m e w ay as ea r-lier, u sin g an y test p oin t X . G en eralizin g, w e see th at ifn 3 is o d d , an d w e are g iv en n p oin ts P 1;P 2;::: ;P n ,th en th ere is ju st on e set o f n p oin ts A 1;A 2;::: ;A n su chth at P i is th e m id p oin t of th e segm en t con n ectin g A i a n dA i+ 1, an d th ese p oin ts m ay b e fo u n d as d escrib ed a b ov e.If n is even , h ow ever, th e situ a tio n is q u ite d i eren t.T ak e th e ca se n = 4; let th e g iv en p oin ts b e P ;Q ;R ;S ,1164 RESONANCE December 2008GENERAL ARTICLEFigure 5.Constructingquadrilateral ABCD fromthe midpoints P, Q, R, S ofitssides: if PQRSisaparal-lelogram, there exist infi-nitely many solutions, oth-erwise there are none.an d let f = H S H R H Q H P . T h en f is a co m p osition ofan ev en n u m b er o f h alf tu rn s, a n d so is a displacem en t. Ifth ere ex ist fou r p oin ts A ;B ;C ;D su ch th a t P ;Q ;R ;S a re,resp ectively, th e m id p oin ts o f B C , C D , D A , A B , th en , a sea rlier, B is a x ed p o in t o f f . B ut a displacem en t w itha xed poin t is the iden tity m ap. S o fo r a solu tio n toex ist w e m u st h av e th e eq u a lity H S H R H Q H P = Id .T h is req u ires th a t !P Q = !R S , i.e., P Q R S m u st b e ap ara llelog ram . If th is is th e ca se, th en for a n y p o in tB , th e q u a d rilateral A B C D w h o se v ertices C ;D ;A ared e n ed b y C = H P (B ), D = H Q (C ), A = H R (D ) is avalid solu tion ; it h a s P ;Q ;R ;S a s th e m id p o in ts of itssid es. S o if on e solu tio n ex ists th en th ere ex ist in n itelym a n y solu tio n s. S ee F igure 5.T h e sam e situ atio n m ain ta in s fo r n = 6; if th e givenp o in ts are P ;Q ;R ;S ;T ;U , n a m ed in cy clic ord er, th ena h ex a gon ex ists w ith th ese p o in ts as th e m id p o in ts ofits sid es if a n d on ly if th e vector su m !P Q + !R S + !T Uvan ish es; an d if th is h o ld s, th en th ere a re in n itely m an ysolu tion s. T h e ca se for n = 8 ;10;1 2;::: is sim ila r.5 . T h e U se O f M a p p in g s In P ro v in g T h e o re m sIsom etries an d sim ilarity m a p s y ield elegan t p ro ofs forcertain g eo m etric resu lts. W e con sid er tw o m em ora b leex a m p les b elow ; th e rst on e w as p o sed a s a p ro b lem inR eson an ce J a n u a ry 20 08 (p p .3 5).Isometries andsimilarity mapsyield elegantproofs for certaingeometric results.1165 RESONANCE December 2008GENERAL ARTICLEFigure 6. Von Aubels theo-rem: PR = QS, PR QS.T h e o re m 4 (V on A u b el). T he cen ters of squ ares draw nextern ally on the sides of a quadrilateral are the verticesof a quadrilateral w hose diagon als are equal an d perpen -dicular to each other.G iven a q u ad rila tera lA B C D , an d sq u ares A B E F , B C G H ,C D IJ , D A K L d raw n ex tern a lly o n its sid es, let th e cen -ters o f th ese sq u a res b e P ;Q ;R ;S , resp ectiv ely (see F ig-ure 6). W e m u st sh ow th at S Q = P R , S Q ? P R .F or th e p ro of, w e u se th e q u arter tu rn s fP ;fQ ;fR ;fSd e n ed b yfP := R P ;9 0 ; fQ:= R Q ;9 0 ; fR := R R ;9 0 ; fS := R S ;9 0 :S in ce th e d isp la cem en t fQ fR fS fPm ap s B b a ckto itself (it is ta ken th rou g h th e cy cle B ;A ;D ;C ;B ), itis th e id en tity m a p . H en ce, th e h alf tu rn s fS fPa n dfQ fRa re id en tical. L et th e cen ter o f ea ch h alf tu rnb e M (see F igure 6). R eca llin g h ow th e cen ter of th eco m p osition o f tw o ro tation s is lo ca ted (T h eo rem 2), w e1166 RESONANCE December 2008GENERAL ARTICLEFigure 7. Napoleons theo-rem: PQR is equilateral.see th at trian gles M P S a n d M Q R are isosceles, w ith arig h t a n gle at M . H en ce, a q u a rter tu rn cen tered at Mtak es S to P , an d Q to R . C on seq u en tly, it ta kes S Q toP R , im p ly in g th at S Q = P R , an d S Q ? P R .T h e o re m 5 . (N ap oleo n ) T he cen ters of equilateral tri-an gles draw n extern ally on the sides of a trian gle are thevertices of an equ ilateral triangle.G iven a tria n gle A B C , an d eq u ilateral tria n gles A B D ,B C E , C A F d raw n ex tern a lly on its sid es, let th e cen tersof th e trian g les b e P ;Q ;R , resp ectively ; see F igure 7.W e m u st sh ow th at trian g le P Q R is eq u ilateral.F or k =1p3 , let f ;g b e com p osite m a p s d e n ed b yf := E (B ;k ) R B ;3 0 ; g := E (C ;k ) R C ;3 0 :E ach m ap is a co m p ositio n of a rota tion th rou g h 30a n dan en largem en t w ith scale fa cto r k (th e tw o rota tio n s areop p ositely d irected ). N ow ob serv e th at: f ta kes D to P , a n d A to R ; h en ce, P R = k D A ,a n d \ (!D A ;!P R ) = 30; g tak es D to P , a n d A to Q ; h en ce, P Q = k D A ,a n d \ (!D A ;!P Q ) = 30.1167 RESONANCE December 2008GENERAL ARTICLEFigure 8. Napoleons theo-rem:PQR is equilateral.H en ce P R = P Q , \ (!P R ;!P Q ) = 6 0, an d th is im p liesth at 4 P Q R is eq u ilateral.R e m a rk 1 . T h ere are m an y fea tu res of in terest in F ig-ure 7; for ex am p le: S egm en ts A D , B E , C F h ave eq u a l len gth , a n d th eycon cu r a t a p oin t T ca lled th e F erm at-T oricellipoin t o f 4 A B C ; T h as th e p rop erty th a t if n o a n -g le of 4 A B C ex ceed s 1 20, th en it is th e p oin t th atm in im izes th e su m o f th e d istan ces to th e verticeso f 4 A B C . S egm en ts A P , B Q , C R con cu r as w ell, at N 1, th e rst N apoleon poin t of 4 A B C . If th e eq u ilateraltrian g les A B D , B C E a n d C A F are d raw n so a sto overla p w ith 4 A B C (rath er th a n lie ou tsid eit), th en A P , B Q , C R co n cu r a t N 2, th e secon dN apoleon poin t of 4 A B C . S ee F igure 8.If no angle oftriangle ABCexceeds 120degree, then T isthe point thatminimizes the sumof the distances tothe vertices of thetriangle.1168 RESONANCE December 2008GENERAL ARTICLEFigure 9. Morleys miracle:PQR is equilateral. F in ally, th e p o in ts o f in tersectio n of (B C ;P Q ), (B C ;P R ), (C A ;Q P ), (C A ; Q R ), (A B ;R P ), (A B ;R Q ) lieo n a co n ic. W e leav e th e p ro of to th e read er. (H in t:U se th e con verse to P ascal's h ex a gon th eorem .)T h e w ell-k n ow n n in e poin t circle theorem m ay also b ep roved in an elega n t m an n er u sin g tran sfo rm a tio n s; b u tw e leave th is to th e read er.6 . M o rle y 's M ira c leW e co n clu d e w ith a recen t p ro of o f on e of th e greatth eorem s of geom etry : M orley's theorem . T h e p ro o f isd u e to th e 198 2 F ield s m ed alist A lain C on n es.T h e re o m 6 (F ra n k M o rley, 1 896 ) T he three poin ts ofin tersectionof the adjacen t trisectors of the an gles ofan y trian gle are the vertices of an equilateral trian gle.S ee F igure 9.T h e th eo rem w as d iscov ered b y F ran k M orley in 18 96,as p a rt of h is w ork on a lg eb ra ic cu rves tan g en t to a givenn u m b er of lin es; h is p ro o f w as h igh ly algeb raic in n a tu re.T h e n icest `p u re geom etry ' p ro o fs are th o se d u e to M TN a ran ien gar (1 909 ) an d J oh n C o n w ay (1 995 ); see [2 ].The well-knownnine point circletheorem may alsobe proved in anelegant mannerusingtransformations.1169 RESONANCE December 2008GENERAL ARTICLEC om p a ct trigo n om etric p ro ofs to o m ay b e fou n d . B u tC on n es's p ro o f is of p a rticu lar in terest a s it u ses g en erala n e m a p p in g s, a n d y ield s a resu lt m ore gen eral th a nM o rley 's th eorem ; see referen ces [3 ],[4 ].L et =23 \ A , =23 \ B , =23 \ C , an d d e n e th erota tion s fa ;fb;fc as follow s: fa = R A ; , fb = R B ; , fc =R C ; . A ssu m e th a t th e vertices of 4 A B C are lab eledin a p ositiv e (`co u n terclo ck w ise') sen se, as sh ow n ; th en(T h eorem 2), P is th e x ed p o in t of R B ; R C ; ; Q is th e x ed p oin t of R C ; R A ; ; R is th e x ed p oin t of R A ; R B ; ; f 3c f 3b f 3a = Id ; fo r, f 3a is eq u ivalen t to re ection in !A B follow ed b y re ection in !B C , a n d sim ilarly forf 3ba n d f 3c . A n eat can cellation n ow ta kes p lace inth e p ro d u ct f 3c f 3b f 3a , giv in g th e d esired relation .W e sh a ll sh ow th at M o rley 's th eorem follow s from th eserelation s an d a gen era l resu lt (T h eorem 7 ) ab ou t a n efu n ction s d e n ed on th e com p lex p lan e C.A n a n e fun ctiong : C ! C is a fu n ctio n of th e ty p eg (x ) = a x + b, w h ere a ;b 2 C, a 6= 0 . If a = 1, th en g is atran slation ; a n d if ja j = 1 b u t a 6= 1 , th en g is a rotation .T h e set of all su ch fu n ction s u n d er com p o sitio n y ield s an on -ab elia n g rou p G . If g (x ) = a x + b is in G , w e d e n e(g ) = a ; th en is a h om om orp h ism fro m G in to th em u ltip lica tive grou p of n on -zero com p lex n u m b ers. F or,if g (x ) = a x + b a n d h (x ) = cx + d are in G , th enh g (x ) = c(a x + b) + d = a cx + (bc + d );) (h g ) = a c = (h ) (g ):T h e k ern el o f is th e su b gro u p o f p u re tran slation s inG . If g (x ) = a x + b is n o t a tran slation , i.e., a 6= 1, th eng h a s a u n iq u e x ed p oin t given b y x (g ) =b1 a :Conness proof isof particularinterest as it usesgeneral affinemappings, andyields a resultmore general thanMorleys theorem.1170 RESONANCE December 2008GENERAL ARTICLET h e o re m 7 (A la in C on n es) L et g1;g2;g3 2G such thatg1 g2, g2 g3, g3 g1, g1 g2 g3 are n ot tran slation s.L et j = (g1 g2 g3), p = x (g1 g2), q = x (g2 g3),r = x (g3 g1). S uppose that g31 g32 g 33= Id ; thenj3= 1 an d p2+ q2+ r2= p q + qr + rp .P roof. L et gk (x ) = ak x + bk . A fter m u ch ca lcu latio nw e g et: g 31 g32 g33 (x ) = a 31a32a 33x + (a21 + a 1 + 1 ) b1 +a31 (a 22 + a2 + 1 ) b2 + (a 1a2)3(a23 + a 3 + 1) b3. H en ce th eeq u a lity g 31 g32 g33 = Id is eq u ivalen t to a 31a 32a33 = 1 a n db = 0 , w h ere b is th e tran slation a l p o rtio n o f g 31 g32 g 33 ,given b y :b = a 21 + a 1 + 1b1 + a31a 22 + a2 + 1b2+(a 1a2)3a 23 + a 3 + 1b3:S in ce j = a 1a 2a3, th e con d itio n a31a32a 33 = 1 is th e sa m eas j3= 1. S h ow in g th at p2+ q2+ r2= p q + qr + rp isted iou s an d u n en lig h ten in g, an d b est left to a com p u teralgeb ra sy stem (see b elow fo r d etails).T h e ap p lica tio n to M o rley 's th eorem is clear: th e h y -p o th eses of th e th eorem h old fo r th e a n e fu n ctio n sfa ;fb;fc, h en ce p2+ q2+ r2= p q + qr + r p . T h is w ell-k n ow n criterion for p ;q;r to b e th e v ertices of an eq u i-la tera l trian gle sh ow s th at 4 P Q R is eq u ila tera l.C on n es's resu lt im p lies th e follow in g tw o stro n ger re-su lts.C o ro lla ry 7 .1 . L et the vertices of 4 A B C be labeledin a positive sen se. F or 0; 0; 02 f 0;1 20;24 0g , let =23 \ A + 0, =23 \ B + 0, =23 \ C + 0, fa=R A ; , fb = R B ; , fc = R C ; . L et P ;Q ;R be, respectively,the xed poin ts of the rotation s R B ; R C ; , R C ; R A ; ,R A ; R B ; . T hen 4 P Q R is equilateral.T h is y ield s a to tal of 18 d i eren t eq u ilateral trian glesasso ciated w ith th e trisecto rs of th e a n gles o f an y tria n -gle. C o rolla ry 7 .2 , w h ich is C on n es's resu lt ex p ressed inan o th er w ay, sh ow s th a t th ere a re eq u ilateral trian glesThis yields a totalof 18 differentequilateraltrianglesassociated with thetrisectors of theangles of anytriangle.1171 RESONANCE December 2008GENERAL ARTICLEasso ciated w ith a n y p a ir of a n e m ap p in gs, su b ject o n lyto som e m ild restrictio n s.C o ro lla ry 7 .2 . L et com plex n um bers a k ;bk (k = 1 ;2)be such that a 1a 2 6= 0 .L et com plex n um bers a3;b3besuch that a 31a32a 33 = 1 an da21 + a 1 + 1b1 + a31a 22 + a 2 + 1b2+(a 1a2)3a 23 + a 3 + 1b3 = 0 :S uppose that a1a2 6= 1, a 2a3 6= 1, a 3a1 6= 1 , a 1a 2a3 6= 1.D e n e p ;q;r as follow s:p =a1b2 + b11 a 1a 2 ; q =a2b3 + b21 a 2a 3 ; r =a 3b1 + b31 a3a 1 :T hen the poin ts correspon din g to p ;q;r are the verticesof an equilateral trian gle.P roof. W e sim p ly o er a M a t h e m a t i c a v eri cation :C l e a r A l l [ a 1 , a 2 , b 1 , b 2 , w , a 3 ,b 3 n , b 3 d , b 3 , p , q , r , z ] ;w = ( - 1 + I S q r t [ 3 ] ) / 2 ;a 3 = w / ( a 1 a 2 ) ;b 3 n = ( a 1 ^ 2 + a 1 + 1 ) b 1 + a 1 ^ 3 ( a 2 ^ 2 + a 2 + 1 ) b 2 ;b r d = a 1 ^ 3 a 2 ^ 3 ( a 3 ^ 2 + a 3 + 1 ) ;b 3 = - b r n / b r d ;p = ( a 1 b 2 + b 1 ) / ( 1 - a 1 a2 ) ;q = ( a 2 b 3 + b 2 ) / ( 1 - a 2 a3 ) ;r = ( a 3 b 1 + b 3 ) / ( 1 - a 3 a1 ) ;z = p ^ 2 + q ^ 2 + r ^ 2 - p q - q r - r p ;F u l l S i m p l i f y [ z ]T h e an sw er is 0; it rem ain s 0 if w e ch an ge th e th ird lin eto a 3 = w ^ 2 / ( a 1 a 2 ) . S in ce z = 0 (w e sh a ll n ot b oth erto a rgu e w ith th e resu lt of aM a t h e m a t i c asim p li ca -tio n ), it fo llow s th at th e p o in ts corresp on d in g to p ;q ;rare in d eed th e vertices of an eq u ilateral trian gle.1172 RESONANCE December 2008GENERAL ARTICLEAddress for CorrespondenceShailesh A ShiraliRishi Valley SchoolRishi Valley 517 352Madanapalle, AP, India.Email:shailesh.shirali@gmail.comSuggested Reading[1] I M Yaglom, Geometric Transformations, Volumes I, II, III, New Math-ematical Library, Math. Assocn America, 1962.[2] A Bogomolny, Morleys Miracle, from Interactive Mathematics Miscel-lany and Puzzles, http://www.cut-the-knot.org/content.shtml[3] AConnes, AViewof Mathematics, available as a pdf file fromhttp://www.alainconnes.org/en/downloads.php[4] A Connes, A new proof of Morleys Theorem, available as a pdf file fromhttp://www.alainconnes.org/en/downloads.phpR e m a rk 2 . C on n es w rites in [4], \T h e p u rp ose of th issh o rt n ote is to giv e a con cep tu al p ro of o f M orley 's th e-orem as a gro u p th eoretic p ro p erty o f th e action of th ea n e g rou p on th e lin e." H e ad d s th a t th e p ro o f m u stm a ke u se of sp ecial E u clid ea n p ro p erties o f th e g rou p ofisom etries, fo r M o rley 's th eorem d o es n ot h old in n o n -E u clid ean g eo m etry.R e m a rk 3 . F ollow in g C on n es's a p p roa ch , on e m ay tryto cast oth er resu lts of p lan e geom etry w ith in a sim ilarfram ew o rk . W e con sid er ju st o n e ex am p le. L et A B C b ean y tria n gle, its v ertices lab eled in a p ositiv e sen se, a n dlet ; ; b e (resp ectively ) th e m easu res of its an glesat A ;B ;C . D e n e th e rotation s fa = R A ; , fb = R B ; ,fc = R C ; ; th en w e h ave: f 2a f 2b f 2c = Id ; th e rota tio n s fa fb, fb fc, fc fa h av e th e sa m e x ed p oin t (i.e., th e sam e cen ter), n am ely, th e in -cen ter I o f 4 A B C ; fa fb fc is a h a lf tu rn cen tered at th e p o in t w h ereth e in circle o f 4 A B C tou ch es sid e C A .