Complex Numbers in Geometry

53 Pages·0.376 MB·English

by  Marko Radovanović

(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:
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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
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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
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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
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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.