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Complex Numbers in Geometry PDF

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(cid:13)c2007TheAuthor(s)andTheIMOCompendiumGroup Complex Numbers in Geometry MarkoRadovanovic´ [email protected] Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 FormulasandTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 ComplexNumbersandVectors. Rotation . . . . . . . . . . . . . . . . . . . . . . . 3 4 TheDistance. RegularPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 PolygonsInscribedinCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 PolygonsCircumscribedAroundCircle . . . . . . . . . . . . . . . . . . . . . . . . 6 7 TheMidpointofArc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 8 ImportantPoints. Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9 Non-uniqueIntersectionsandViete’sformulas. . . . . . . . . . . . . . . . . . . . . 8 10 DifferentProblems–DifferentMethods . . . . . . . . . . . . . . . . . . . . . . . . 8 11 DisadvantagesoftheComplexNumberMethod . . . . . . . . . . . . . . . . . . . . 10 12 HintsandSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 13 ProblemsforIndepentStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1 Introduction When we are unable to solve some problem in plane geometry, it is recommended to try to do calculus. Thereareseveraltechniquesfordoingcalculationsinsteadofgeometry. Thenexttextis devotedtooneofthem–theapplicationofcomplexnumbers. The plane will be the complex plane and each point has its corresponding complex number. Becauseofthatpointswillbeoftendenotedbylowercaselettersa,b,c,d,...,ascomplexnumbers. Thefollowingformulascanbederivedeasily. 2 FormulasandTheorems a b c d Theorem1. ab cdifandonlyif − = − . • k a b c d − − a b a c a,b,carecolinearifandonlyif − = − . • a b a c − − a b c d ab cdifandonlyif − = − . • ⊥ a b −c d − − c b c a ϕ=∠acb(fromatobinpositivedirection)ifandonlyif − =eiϕ − . • c b c a | − | | − | Theorem2. Propertiesoftheunitcircle: 2 OlympiadTrainingMaterials,www.imo.org.yu,www.imocompendium.com a b Forachordabwehave − = ab. • a b − − a+b c Ifcbelongstothechordabthenc = − . • ab 2ab Theintersectionofthetangentsfromaandbisthepoint . • a+b 1 Thefootofperpendicularfrom anarbitrarypointcto thechordabisthepointp = a+ • 2 b+c abc . (cid:16) − (cid:17) ab(c+d) cd(a+b) Theintersectionofchordsabandcdisthepoint − . • ab cd − Theorem3. Thepointsa,b,c,dbelongtoacircleifandonlyif a c a d − : − R. b c b d ∈ − − Theorem4. Thetrianglesabcandpqraresimilarandequallyorientedifandonlyif a c p r − = − . b c q r − − Theorem5. Theareaofthetriangleabcis a a 1 i i p= b b 1 = ab +bc +ca ab bc ca. 4(cid:12) (cid:12) 4 − − − (cid:12) c c 1 (cid:12) (cid:12) (cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) a+λb Theorem6. Thepo(cid:12)intcdivides(cid:12)thesegmentabintheratioλ= 1ifandonlyifc= . • 6 − 1+λ a+b+c Thepointtisthecentroidofthetriangleabcifandonlyift= . • 3 Fortheorthocenterhandthecircumcenteroofthetriangleabcwehaveh+2o=a+b+c. • Theorem7. Supposethattheunitcircleisinscribedinatriangleabcandthatittouchesthesides bc,ca,ab,respectivelyatp,q,r. 2qr 2rp 2pq Itholdsa= ,b= andc= ; • q+r r+p p+q Fortheorthocenterhofthetriangleabcitholds • 2(p2q2+q2r2+r2p2+pqr(p+q+r)) h= . (p+q)(q+r)(r+p) 2pqr(p+q+r) Fortheexcenteroofthetriangleabcitholdso= . • (p+q)(q+r)(r+p) Theorem8. Foreach triangleabc inscribed in a unitcircle there are numbersu,v,w such • thata = u2,b = v2,c = w2, and uv, vw, wu are the midpointsofthe arcs ab,bc,ca − − − (respectively)thatdon’tcontainc,a,b. Fortheabovementionedtriangleanditsincenteriwehavei= (uv+vw+wu). • − MarkoRadovanovic´:ComplexNumbersinGeometry 3 Theorem9. Considerthetriangle whoseonevertexis0,andtheremainingtwoarexandy. △ (xy+xy)(x y) Ifhistheorthocenterof thenh= − . • △ xy xy − xy(x y) Ifoisthecircumcenterof ,theno= − . • △ xy xy − 3 ComplexNumbers and Vectors. Rotation This section contains the problems that use the main properties of the interpretation of complex numbers as vectors (Theorem 6) and consequences of the last part of theorem 1. Namely, if the pointbisobtainedbyrotationofthepointaaroundcfortheangleϕ(inthepositivedirection),then b c=eiϕ(a c). − − 1. (YugMO1990,3-4grade)LetS bethecircumcenterandH theorthocenterof ABC. LetQ △ be the point such that S bisects HQ and denote by T , T , and T , respectively, the centroids of 1 2 3 BCQ, CAQand ABQ. Provethat △ △ △ 4 AT =BT =CT = R, 1 2 3 3 whereRdenotesthecircumradiusof ABC. △ 2. (BMO1984)LetABCD beaninscribedquadrilateralandletH ,H ,H andH betheor- A B C D thocentersofthetrianglesBCD,CDA,DAB,andABCrespectively.Provethatthequadrilaterals ABCDandH H H H arecongruent. A B C D 3. (YugTST1992)ThesquaresBCDE, CAFG, andABHI areconstructedoutsidethetriangle ABC. LetGCDQandEBHP beparallelograms.Provethat APQisisoscelesandrectangular. △ 4. (YugMO1993,3-4grade)TheequilateraltrianglesBCB ,CDC ,andDAD areconstructed 1 1 1 outsidethetriangleABC. IfP andQarerespectivelythemidpointsofB C andC D andifRis 1 1 1 1 themidpointofAB,provethat PQRisisosceles. △ 5. Inthe planeofthe triangleA A A the pointP isgiven. DenotewithA = A , forevery 1 2 3 0 s s−3 naturalnumbers > 3. ThesequenceofpointsP ,P ,P ,...isconstructedinsuchawaythatthe 0 1 2 pointP isobtainedbytherotationofthepointP foranangle120o intheclockwisedirection k+1 k aroundthepointA . ProvethatifP =P ,thenthetriangleA A A hastobeisosceles. k+1 1986 0 1 2 3 6. (IMOShortlist1992)LetABCD be a convexquadrilateralforwhichAC = BD. Equilateral trianglesare constructedon the sides of the quadrilateral. Let O , O , O , and O be the centers 1 2 3 4 ofthetrianglesconstructedonAB,BC,CD,andDArespectively. ProvethatthelinesO O and 1 3 O O areperpendicular. 2 4 4 The Distance. RegularPolygons Inthissectionwewillusethefollowingbasicrelationforcomplexnumbers: a2 =aa. Similarly, | | for calculating the sums of distances it is of great advantageif points are colinear or on mutually parallel lines. Hence it is often very useful to use rotations that will move some points in nice positions. Now we will consider the regular polygons. It is well-known that the equation xn = 1 has exactlynsolutionsincomplexnumbersandtheyareoftheformxk = ei2knπ,for0 k n 1. ≤ ≤ − Nowwehavethatx =1andx =εk,for1 k n 1,wherex =ε. 0 k 1 ≤ ≤ − Let’slookatthefollowingexamplefortheillustration: Problem1. LetA A A A A A A bearegular7-gon.Provethat 0 1 2 3 4 5 6 1 1 1 = + . A A A A A A 0 1 0 2 0 3 4 OlympiadTrainingMaterials,www.imo.org.yu,www.imocompendium.com Solution. As mentionedabovelet’s take ak = εk, for 0 k 6, where ε = ei27π. Further, by rotation around a0 = 1 for the angle ε, i.e. ω = ei21π4,≤the p≤oints a1 and a2 are mapped to a′ and a′ respectively. These two points are collinear with a . Now it is enough to prove that 1 2 3 1 1 1 = + . Since ε = ω2, a′ = ε(a 1)+1, and a′ = ω(a 1)+1 it is a′ 1 a′ 1 a 1 1 1 − 2 2 − 1− 2− 3− enoughtoprovethat 1 1 1 = + . ω2(ω2 1) ω(ω4 1) ω6 1 − − − After rearranging we get ω6 +ω4 +ω2 +1 = ω5 +ω3 +ω. From ω5 = ω12, ω3 = ω10, − − and ω = ω8 (which can be easily seen from the unit circle), the equality follows from 0 = − ε7 1 ω12+ω10+ω8+ω6+ω4+ω2+1=ε6+ε5+ε4+ε3+ε2+ε+1= − =0. ε 1 △ − 7. LetA A ...A bearegular15-gon.Provethat 0 1 14 1 1 1 1 = + + . A A A A A A A A 0 1 0 2 0 4 0 7 8. LetA A ...A be a regularn-goninscribed in a circle with radiusr. Provethatfor every 0 1 n−1 pointP ofthecircleandeverynaturalnumberm<nwehave n−1 2m PA2m = nr2m. k m k=0 (cid:18) (cid:19) X 9. (SMN TST2003)LetM andN be two differentpointsin the planeof thetriangleABC such that AM :BM :CM =AN :BN :CN. ProvethatthelineMN containsthecircumcenterof ABC. △ 10. Let P be an arbitrary point on the shorter arc A A of the circle circumscribed about the 0 n−1 regularpolygonA A ...A .Leth ,h ,...,h bethedistancesofP fromthelinesthatcontain 0 1 n−1 1 2 n theedgesA A ,A A ,...,A A respectively.Provethat 0 1 1 2 n−1 0 1 1 1 1 + + + = . h h ··· h h 1 2 n−1 n 5 PolygonsInscribed inCircle Intheproblemswherethepolygonisinscribedinthecircle,itisoftenusefultoassumethattheunit circleisthecircumcircleofthepolygon.Intheorem2wecanseelotofadvantagesoftheunitcircle (especially the first statement) and in practice we will see that lot of the problems can be solved usingthis method. Inparticular,we knowthateachtriangleis inscribedin thecircle andin many problemsfromthegeometryoftrianglewecanmakeuseofcomplexnumbers.Theonlyproblemin thistaskisfindingthecircumcenter.Forthatyoushouldtakealookinthenexttwosections. 11. ThequadrilateralABCD isinscribedinthe circlewithdiameterAC. ThelinesAB andCD intersectatM andthetangetstothecircleatBandC intersetatN. ProvethatMN AC. ⊥ 12.(IMOShorlist1996)LetH betheorthocenterofthetriangle ABCandP anarbitrarypointof △ itscircumcircle.LetEthefootofperpendicularBHandletPAQBandPARCbeparallelograms. IfAQandHRintersectinX provethatEX AP. k 13.GivenacyclicquadrilateralABCD,denotebyP andQthepointssymmetrictoC withrespect toABandADrespectively.ProvethatthelinePQpassesthroughtheorthocenterof ABD. △ MarkoRadovanovic´:ComplexNumbersinGeometry 5 14. (IMO Shortlist 1998) Let ABC be a triangle, H its orthocenter, O its incenter, and R the circumradius. LetD bethepointsymmetrictoAwithrespecttoBC,E thepointsymmetrictoB withrespecttoCA, andF thepointsymmetrictoC withrespecttoAB. ProvethatthepointsD, E,andF arecollinearifandonlyifOH =2R. 15. (Rehearsal Competition in MG 2004) Given a triangle ABC, let the tangent at A to the cir- cumscribedcircleintersectthemidsegmentparalleltoBC atthepointA . Similarlywedefinethe 1 pointsB andC . ProvethatthepointsA ,B ,C lieonalinewhichisparalleltotheEulerlineof 1 1 1 1 1 ABC. △ 16. (MOP 1995) Let AA and BB be the altitudes of ABC and let AB = AC. If M is the 1 1 △ 6 midpointofBC,H theorthocenterof ABC,andDtheintersectionofBC andB C ,provethat 1 1 △ DH AM. ⊥ 17. (IMOShortlist1996)LetABC beanacute-angledtrianglesuchthatBC > CA. LetObethe circumcircle,H theorthocenter,andF thefootofperpendicularCH. IftheperpendicularfromF toOF intersectsCAatP,provethat∠FHP =∠BAC. 18.(Romania2005)LetA A A A A A beaconvexhexagoninscribedinacircle.LetA′,A′,A′ 0 1 2 3 4 5 0 2 4 bethepointsonthatcirclesuchthat A A′ A A , A A′ A A A A′ A A . 0 0 k 2 4 2 2 k 4 0 4 4 k 2 0 Supposethatthe linesA′A andA A intersectatA′, thelinesA′A andA A intersectatA′, 0 3 2 4 3 2 5 0 4 5 andthelinesA′A andA A intersectatA′. 4 1 0 2 1 IfthelinesA A ,A A ,andA A areconcurrent,provethatthelinesA A′,A A′ andA A′ are 0 3 1 4 2 5 0 3 4 1 2 5 concurrentaswell. 19.(Simson’sline)IfA,B,Carepointsonacircle,thenthefeetofperpendicularsfromanarbitrary pointDofthatcircletothesidesofABC arecollinear. 20. Let A, B, C, D be four points on a circle. Prove that the intersection of the Simsons line correspondingtoAwithrespecttothetriangleBCDandtheSimsonslinecorrespondingtoBw.r.t. ACDbelongstothelinepassingthroughC andtheorthocenterof ABD. △ △ 21.Denotebyl(S;PQR)theSimsonslinecorrespondingtothepointS withrespecttothetriangle PQR. If the points A,B,C,D belong to a circle, prove that the lines l(A;BCD), l(B;CDA), l(C,DAB),andl(D,ABC)areconcurrent. 22. (Taiwan 2002) Let A, B, and C be fixed points in the plane, and D the mobile point of the circumcircle of ABC. Let I denote the Simsons line of the point A with respect to BCD. A △ △ SimilarlywedefineI ,I ,andI . FindthelocusofthepointsofintersectionofthelinesI ,I , B C D A B I ,andI whenDmovesalongthecircle. C D 23. (BMO2003)GivenatriangleABC, assumethatAB = AC. LetD betheintersectionofthe 6 tangent to the circumcircle of ABC at A with the line BC. Let E and F be the points on the △ bisectorsofthesegmentsABandAC respectivelysuchthatBE andCF areperpendiculartoBC. ProvethatthepointsD,E,andF lieonaline. 24.(Pascal’sTheorem)IfthehexagonABCDEF canbeinscribedinacircle,provethatthepoints AB DE,BC EF,andCD FAarecolinear. ∩ ∩ ∩ 25. (Brokard’sTheorem)LetABCD beaninscribedquadrilateral. ThelinesAB andCD inter- sectatE,thelinesADandBC intersectinF,andthelinesAC andBDintersectinG. Provethat OistheorthocenterofthetriangleEFG. 26. (Iran2005)LetABC beanequilateraltrianglesuchthatAB =AC. LetP bethepointonthe extentionofthesideBC andletX andY bethepointsonABandAC suchthat PX AC, PY AB. k k 6 OlympiadTrainingMaterials,www.imo.org.yu,www.imocompendium.com LetT bethemidpointofthearcBC. ProvethatPT XY. ⊥ 27.LetABCDbeaninscribedquadrilateralandletK,L,M,andN bethemidpointsofAB,BC, CA, and DA respectively. Prove that the orthocenters of AKN, BKL, CLM, DMN △ △ △ △ formaparallelogram. 6 PolygonsCircumscribed Around Circle Similarly as in the previous chapter, here we will assume that the unit circle is the one inscribed in the givenpolygon. Againwe will make a use of theorem2 and especially its third part. In the case of triangle we use also the formulas from the theorem 7. Notice that in this case we know boththeincenterandcircumcenterwhichwasnotthecaseintheprevioussection. Also,noticethat the formulas from the theorem 7 are quite complicated, so it is highly recommended to have the circumcircleforastheunitcirclewheneverpossible. 28. ThecirclewiththecenterOisinscribedinthetriangleABC andittouchesthesidesAB,BC, CAinM,K,Erespectively.DenotebyP theintersectionofMKandAC. ProvethatOP BE. ⊥ 29. ThecirclewithcenterOisinscribedinaquadrilateralABCDandtouchesthesidesAB,BC, CD, andDArespectivelyinK,L, M, andN. ThelinesKLandMN intersectatS. Provethat OS BD. ⊥ 30. (BMO 2005) Let ABC be an acute-angled triangle which incircle touches the sides AB and AC inD andE respectively. LetX andY betheintersectionpointsofthebisectorsoftheangles ∠ACB and∠ABC withthelineDE. LetZ bethemidpointofBC. ProvethatthetriangleXYZ isisoscelesifandonlyif∠A=60◦. 31. (NewtonsTheorem)Givenan circumscribedquadrilateralABCD, letM andN bethemid- pointsofthediagonalsAC andBD. IfS istheincenter,provethatM,N,andS arecolinear. 32. LetABCD bea quadrilateralwhoseincircletouchesthesidesAB, BC, CD, andDAatthe pointsM,N,P,andQ. ProvethatthelinesAC,BD,MP,andNQareconcurrent. 33. (Iran1995)Theincircleof ABC touchesthesidesBC, CA, andAB respectivelyinD,E, △ andF. X,Y,andZ arethemidpointsofEF,FD,andDE respectively.Provethattheincenterof ABC belongstothelineconnectingthecircumcentersof XYZ and ABC. △ △ △ 34.AssumethatthecirclewithcenterI touchesthesidesBC,CA,andABof ABCinthepoints △ D,E,F,respectively.AssumethatthelinesAI andEF intersectatK,thelinesEDandKC atL, andthelinesDF andKBatM. ProvethatLM isparalleltoBC. 35. (25. TournamentofTowns)GivenatriangleABC,denotebyH itsorthocenter,I theincenter, Oitscircumcenter,andK thepointoftangencyofBC andtheincircle.IfthelinesIOandBC are parallel,provethatAOandHK areparallel. 36. (IMO2000)LetAH ,BH ,andCH bethealtitudesoftheacute-angledtriangleABC. The 1 2 3 incircleofABC touchesthesidesBC, CA, AB respectivelyinT , T , andT . Letl , l , andl 1 2 3 1 2 3 bethelinessymmetrictoH H ,H H ,H H withrespecttoT T ,T T ,andT T respectively. 2 3 3 1 1 2 2 3 3 1 1 2 Provethatthelinesl ,l ,l determineatriagnlewhoseverticesbelongtotheincircleofABC. 1 2 3 7 The MidpointofArc Weoftenencounterproblemsinwhichsomepointisdefinedtobethemidpointofanarc.Oneofthe difficultiesinusingcomplexnumbersisdistinguishingthearcsofthecirle.Namely,ifwedefinethe midpointofanarctobetheintersectionofthebisectorofthecorrespondingchordwiththecircle, we are getting two solutions. Such problems can be relatively easy solved using the first part of thetheorem8. Moreoverthesecondpartofthetheorem8givesanalternativewayforsolvingthe problemswith incirclesand circumcircles. Notice thatthe coordinatesof the importantpointsare MarkoRadovanovic´:ComplexNumbersinGeometry 7 givenwiththeequationsthataremuchsimplerthanthoseintheprevioussection. Howeverwehave a problem when calculating the points d,e,f of tangency of the incircle with the sides (calculate them!),sointhiscaseweusethemethodsoftheprevioussection. Inthecaseofthenon-triangular polygonwealsoprefertheprevioussection. 37. (KvantM769) Let L be the incenter of the triangle ABC and let the lines AL, BL, and CL intersectthecircumcircleof ABC atA ,B ,andC respectively.LetRbethecircumradiusand 1 1 1 △ rtheinradius.Provethat: LA LC LA LB S(ABC) 2r (a) 1· 1 =R; (b) · =2r; (c) = . LB LC S(A B C ) R 1 1 1 1 38. (Kvant M860) Let O and R be respectively the center and radius of the circumcircle of the triangleABC andletZ and r berespectivelythe incenterandinradiusof ABC. DenotebyK △ thecentroidofthetriangleformedbythepointsoftangencyoftheincircleandthesides. Provethat Z belongstothesegmentOK andthatOZ :ZK =3R/r. 39. LetP betheintersectionofthediagonalsAC andBD oftheconvexquadrilateralABCD for whichAB =AC =BD. LetOandI bethecircumcenterandincenterofthetriangleABP. Prove thatifO =I thenOI CD. 6 ⊥ 40. LetI betheincenterofthetriangleABC forwhichAB =AC. LetO bethepointsymmetric 1 6 tothecircumcenterof ABC withrespecttoBC. ProvethatthepointsA,I,O arecolinearifand 1 onlyif∠A=60◦. △ 41. GivenatriangleABC,letA ,B ,andC bethemidpointsofBC,CA,andABrespecctively. 1 1 1 LetP,Q,andRbethepointsoftangencyoftheincirclekwiththesidesBC,CA,andAB. LetP , 1 Q ,andR bethemidpointsofthearcsQR,RP,andPQonwhichthepointsP,Q,andRdivide 1 1 thecirclek,andletP ,Q ,andR bethemidpointsofarcsQPR,RQP,andPRQrespectively. 2 2 2 ProvethatthelinesA P ,B Q , andC R areconcurrent,aswellasthelinesA P , B Q ,and 1 1 1 1 1 1 1 1 1 2 C R . 1 2 8 ImportantPoints. Quadrilaterals In the last three sections the points that we’ve taken as initial, i.e. those with known coordinates have been ”equally improtant” i.e. all of them had the same properties (they’ve been either the pointsofthe same circle, orintersectionsof the tangentsof the same circle, etc.). However,there are numerous problems where it is possible to distinguish one point from the others based on its influenceto the other points. Thatpointwill be regardedas the origin. This is particularlyuseful in the case of quadrilaterals (that can’t be inscribed or circumscribed around the circle) – in that casetheintersectionofthediagonalscanbeagoodchoicefortheorigin. Wewillmakeuseofthe formulasfromthetheorem9. 42.ThesquaresABB′B′′,ACC′C′′,BCXY areconsctructedintheexteriorofthetriangleABC. LetP bethecenterofthesquareBCXY. ProvethatthelinesCB′′,BC′′,AP intersectinapoint. 43. LetO betheintersectionofdiagonalsofthequadrilateralABCD andM,N themidpointsof thesideAB andCD respectively. ProvethatifOM CD andON AB thenthequadrilateral ⊥ ⊥ ABCDiscyclic. 44. LetF bethepointonthebaseABofthetrapezoidABCDsuchthatDF =CF. LetE bethe intersectionofAC andBDandO andO thecircumcentersof ADF and FBC respectively. 1 2 △ △ ProvethatFE O O . 1 2 ⊥ 45. (IMO2005)LetABCDbeaconvexquadrilateralwhosesidesBC andADareofequallength but not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The lines AC and BD intersect at P, the lines BD and EF intersectat Q, and the 8 OlympiadTrainingMaterials,www.imo.org.yu,www.imocompendium.com linesEF andAC intersectatR. ConsiderallsuchtrianglesPQRasE andF vary. Showthatthe circumcirclesofthesetriangleshaveacommonpointotherthanP. 46. Assume that the diagonals of ABCD intersect in O. Let T and T be the centroids of the 1 2 trianglesAODandBOC,andH andH orthocentersof AOBand COD. ProvethatT T 1 2 1 2 △ △ ⊥ H H . 1 2 9 Non-unique Intersections and Viete’s formulas Thepointofintersectionoftwolinescanbedeterminedfromthesystemoftwoequationseachof whichcorrespondstotheconditionthatapointcorrespondtoaline. Howeverthismethodcanlead usintosomedifficulties. Aswementionedbeforestandardmethodscanleadtonon-uniquepoints. Forexample,ifwewanttodeterminetheintersectionoftwocircleswewillgetaquadraticequations. Thatisnotsurprisingatallsincethetwocircleshave,ingeneral,twointersectionpoints. Also, in manyoftheproblemswedon’tneedbothofthese points,justthedirectionofthe linedetermined bythem. Similarly,wemayalreadyknowoneofthepoints. Inbothcasesitismoreconvenientto useVieta’sformulasandgetthesumsandproductsofthesepoints. Thuswecanavoid”takingthe squarerootofacomplexnumber”whichisverysuspiciousoperationbyitself,andusuallyrequires someknowledgeofcomplexanalysis. Letusmakearemark: Ifweneedexplicitlycoordinatesofoneoftheintersectionpointsoftwo circles,andwedon’tknowtheother,theonlywaytosolvethisproblemusingcomplexnumbersis tosetthegivenpointtobeoneoftheinitialpoints. 47. SupposethatthetangentstothecircleΓatAandB intersectatC. ThecircleΓ whichpasses 1 through C and touches AB at B intersects the circle Γ at the point M. Prove that the line AM bisectsthesegmentBC. 48. (Republic Competition 2004, 3rd grade) Given a circle k with the diameter AB, let P be an arbitrarypointofthecircledifferentfromAandB. TheprojectionsofthepointP toABisQ. The circlewith thecenterP andradiusPQintersectsk atC andD. LetE betheintersectionofCD andPQ. LetF bethemidpointofAQ,andGthefootofperpendicularfromF toCD. Provethat EP =EQ=EGandthatA,G,andP arecolinear. 49. (China1996)LetH betheorthocenterofthetriangleABC. ThetangentsfromAtothecircle withthediameterBC intersectthecircleatthepointsP andQ. ProvethatthepointsP,Q,andH arecolinear. 50. LetP bethepointontheextensionofthediagonalAC oftherectangleABCD overthepoint C suchthat∠BPD =∠CBP. DeterminetheratioPB :PC. 51. (IMO2004)IntheconvexquadrilateralABCD thediagonalBD isnotthebisectorofanyof theanglesABC andCDA. LetP bethepointintheinteriorofABCDsuchthat ∠PBC =∠DBAand∠PDC =∠BDA. ProvethatthequadrilateralABCDiscyclicifandonlyifAP =CP. 10 Different Problems – Different Methods Inthissectionyouwillfindtheproblemsthatarenotcloselyrelatedtosomeofthepreviouschapters, as well as the problemsthat are related to more than one of the chapters. The useful advice is to carefully thinkof possible initial points, the origin, and the unitcircle. As you will see, the main problemwith solvingthese problemsis the time. Thusif youare in competitionand you want to usecomplexnumbersitisveryimportantforyoutoestimatethetimeyouwillspend. Havingthis inmind,itisveryimportanttolearncomplexnumbersasearlyaspossible. Youwillseeseveralproblemsthatusetheorems3,4,and5. MarkoRadovanovic´:ComplexNumbersinGeometry 9 52. Given four circles k , k , k , k , assume that k k = A ,B , k k = A ,B , 1 2 3 4 1 2 1 1 2 3 2 2 ∩ { } ∩ { } k k = A ,B ,k k = A ,B . IfthepointsA ,A ,A ,A lieonacircleoronaline, 3 4 3 3 4 1 4 4 1 2 3 4 ∩ { } ∩ { } provethatthepointsB ,B ,B ,B lieonacircleoronaline. 1 2 3 4 53. Suppose that ABCD is a parallelogram. The similar and equally oliented triangles CD and CB areconstructedoutsidethisparallelogram. ProvethatthetriangleFAE issimilarandequally orientedwiththefirsttwo. 54. ThreetrianglesKPQ,QLP,andPQM areconstructedonthesamesideofthesegmentPQ insuchawaythat∠QPM = ∠PQL= α,∠PQM = ∠QPK = β,and∠PQK = ∠QPL= γ. Ifα<β <γ andα+β+γ =180◦,provethatthetriangleKLM issimilartothefirstthree. 55. ∗(Iran, 2005) Let n be a prime number and H a convex n-gon. The polygons H ,...,H 1 2 n are definedrecurrently: theverticesofthe polygonH areobtainedfromtheverticesofH by k+1 k symmetrythroughk-thneighbour(inthepositivedirection).ProvethatH andH aresimilar. 1 n 56. Provethatthearea ofthetriangleswhoseverticesarefeetofperpendicularsfromanarbitrary vertexofthecyclicpentagontoitsedgesdoesn’tdependonthechoiceofthevertex. 57.ThepointsA ,B ,C arechoseninsidethetriangleABC tobelongtothealtitudesfromA,B, 1 1 1 C respectively.If S(ABC )+S(BCA )+S(CAB )=S(ABC), 1 1 1 provethatthequadrilateralA B C H iscyclic. 1 1 1 58. (IMOShortlist1997)ThefeetofperpendicularsfromtheverticesA, B, andC ofthetriangle ABC are D, E, end F respectively. The line through D parallel to EF intersects AC and AB respectively in Q and R. The line EF intersects BC in P. Prove that the circumcircle of the trianglePQRcontainsthemidpointofBC. 59. (BMO 2004) Let O be a point in the interior of the acute-angled triangle ABC. The circles throughOwhosecentersarethemidpointsoftheedgesof ABC mutuallyintersectatK,L,and △ M, (differentfrom O). Prove that O is the incenter of the triangle KLM if and only if O is the circumcenterofthetriangleABC. 60. Twocirclesk andk aregivenintheplane. LetAbetheircommonpoint. Twomobilepoints, 1 2 M and M move along the circles with the constant speeds. They pass throughA always at the 1 2 sametime. ProvethatthereisafixedpointP thatisalwaysequidistantfromthepointsM andM . 1 2 61. (Yug TST 2004) Given the square ABCD, let γ be i circle with diameter AB. Let P be an arbitrarypointonCD, andletM andN beintersectionsofthelinesAP andBP withγ thatare differentfromA and B. Let Q be the pointof intersectionof the linesDM and CN. Provethat Q γ andAQ:QB =DP :PC. ∈ 62. (IMOShortlist1995)GiventhetriangleABC,thecirclepassingthroughB andC intersectthe sides AB andAC againin C′ and B′ respectively. Provethat the lines BB′, CC′, and HH′ are concurrent,whereH andH′orthocentersofthetrianglesABC andA′B′C′ respectively. 63. (IMOShortlist1998)LetM andN beinteriorpointsofthetriangleABC suchthat∠MAB = ∠NAC and∠MBA=∠NBC.Provethat AM AN BM BN CM CN · + · + · =1. AB AC BA BC CA CB · · · 64. (IMOShortlist1998)LetABCDEF beaconvexhexagonsuchthat∠B+∠D+∠F =360◦ andAB CD EF =BC DE FA.Provethat · · · · BC AE FD =CA EF DB. · · · · 65. (IMOShortlist1998)LetABC beatrianglesuchthat∠A = 90◦and∠B < ∠C. Thetangent atAtoitscircumcircleω intersectthe lineBC atD. LetE bethereflectionofAwithrespectto 10 OlympiadTrainingMaterials,www.imo.org.yu,www.imocompendium.com BC, X the footof the perpendicularfrom A to BE, and Y the midpointof AX. If the line BY intersectsωinZ,provethatthelineBDtangentsthecircumcircleof ADZ. △ Hint: Usesomeinversionfirst... 66. (RehearsalCompetitioninMG1997,3-4grade)GivenatriangleABC,thepointsA ,B and 1 1 C arelocatedonitsedgesBC,CA,andAB respectively. Supposethat ABC A B C . If 1 1 1 1 △ ∼△ either the orthocentersor the incentersof ABC and A B C coincideprovethatthe triangle 1 1 1 △ △ ABC isequilateral. 67.(Ptolomy’sinequality)ProvethatforeveryconvexquadrilateralABCDthefollowinginequal- ityholds AB CD+BC AD AC BD. · · ≥ · 68. (China1998)FindthelocusofallpointsDsuchthat DA DB AB+DB DC BC+DC DA CA=AB BC CA. · · · · · · · · 11 Disadvantages oftheComplex Number Method Thebigestdifficultiesintheuseofthemethodofcomplexnumberscanbeencounteredwhendealing withtheintersectionofthelines(aswecanseefromthefifthpartofthetheorem2,althoughitdealt withthechordsofthecircle).Also,thedifficultiesmayarrisewhenwehavemorethanonecirclein theproblem. Henceyoushouldavoidusingthecomplesnumbersinproblemswhentherearelotof linesingeneralpositionwithoutsomespecialcircle,orwhentherearemorethentwocircles. Also, thethingscangetverycomplicatedifwehaveonlytwocirclesingeneralposition,andonlyinthe rarecasesyouareadvisedtousecomplexnumbersinsuchsituations. Theproblemswhensomeof theconditionsistheequlitywithsumsofdistancesbetweennon-colinearpointscanbeverydifficult andpretty-muchunsolvablewiththismethod. Ofcourse,theseareonlytheobvioussituationswhenyoucan’tcountonhelpofcomplexnum- bers. Therearenumerousinnocent-lookingproblemswherethecalculationcangiveusincreadible difficulties.

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