111 Nice Geometry Problems

8/12/2019 111 Nice Geometry Problems

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Nice Geometry Problems

Dedicated to Mathscopers

Problem 1:(O1), (O2)are the B , Cmixtilinear excircles ofABC.(I)is the incenter ofABC,

it touches AB, AC,BC at K, L, F resp. CK cuts (O2) at M, BL cuts (O1) at N, M B cuts CN at

P. Prove thatBAF =

CAP.

Problem 2: Let ABC be a triangle inscribed in circumcircle (O). Denote A1, B1.C1 respectively

to be the projections ofA, B , C on to BC,CA,AB. Let A2, B2, C2 respectively be the intersections

ofAO, BO, CO with BC,CA,AB. A circle a passes through A1, A2 and lies tangent to the arc of

BCthat does not contain A of(O) at Ta. The same denition holds for Tb, Tc. Prove that ATa, BTb

and CTc are concurrent.

Problem 3: Let ABCbe a triangle. Point M and N lie on sides AC and BCrespectively such

that M N||AB. PointsP andQ lie on sides AB andC Brespectively such that P Q||AC. The incircle

of triangle CM N touches segment AC at E. The incircle of triangle BP Q touches segment AB at

F. LineE Nand AB meet at R, and lines F Qand ACmeet at S. Given that AE=AF, prove that

the incenter of triangle AEF lies on the incircle of triangle ARS.

Problem 4: Given three parallel lines l1, l2, l3, points A, B, Care on l1, l2, l3, respectively AC

intersect l2 at point N, angle ABC = 90 point D is on line l2 such that N is midpoint ofBD, let

circles a, b wich goes through point D such that a is tangent to l1, b is tangent to l3, a, b intersects

at point E, J is midpoint ofDE. Prove that line BJgoes through midpoint ofAC.

Problem 5:BD, CEare the altitudes ofABC.Mis the midpoint ofB C.BD, CEcut(EDM)

at S, R resp. SC cuts BR at Q. ES cuts DR at G. Prove that A, Q, G are collinear.

Problem 6: Given triangle AB Cwith its circumcircle (O)and its orthocenter H. Let HaHbHc be

the orthic triangle of triangle ABC.P is an arbitrary point on Euler line. AP, BP, CP intersect (O)

again atA1, B1, C1.Let A2, B2, C2 be the reflections ofA1, B1, C1wrt Ha, Hb, Hc, respectively. Prove

that H, A2, B2, C2 are concyclic.

Problem 7: A triangle ABC is given with orthocenter H and circumcenter O and let P be,

an abitrary point on OH. We denote the points K (O) BP, L (O) CP, where (O) is the

circumcircle ofABC and let be the points X, Z, as the symmetric points ofK, L respectively,

with respect to the points E AC BH and F AB CH. The lines through the points X, Z

and perpendicular to HX, HZ, intersect the line segments KH, LH respectively, at points M, N.

Prove that M NK L.

Problem 8: O1 and O2 are mixtillnear excircles, they cut lines AB,BC and CA at pointsD , E , F , Grespectively. PointsM , Nare the midpoints ofDEandGF, connectMto Fand connect

N F, M F O1O2= W, NE O1O2=V. Prove that W B

V C =

AB

CA

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Problem 9: Let ABC be a triangle. Let be the inscribed circle. A, B, C are the points of

tangency on the side BC,AC,AB. The point Tis the intersection between the extension ofB Cand

CB. Let Q be the intersection ofAA and the inscribed circle . Show that T Q is tangent to .

Problem 10: The A mixtilinear incircle of ABC cuts (ABC) at P. D is on BC such that

AC+ BD = AB + CD. Prove that DP C= ACB.

Problem 11: ABCa triangle. P: a point inside ABC.1, 2, 3: the resp. circumcircles of the tri-

angles PBC , PC A , PA B. U the midpoint of the arc BP C. V, W the resp. midpoints of the arcs

CA,AB which doesnt contain P. Prove that: P , U, V , W are concyclic.

Problem 12: Let ABCbe an acute-angled triangle, and let P and Q be two points on its side

BC. Construct a point C1 in such a way that the convex quadrilateral AP BC1 is cyclic, QC1C A,

and the points C1 and Q lie on opposite sides of the line AB. Construct a point B1 in such a way

that the convex quadrilateral AP CB1 is cyclic, QB1B A, and the points B1 and Qlie on oppositesides of the line AC. Prove that the points B1, C1, P, and Qlie on a circle.

Problem 13: Let(P)and (Q)be two circles on a plane such that (Q)lies inside(P).Three circles

A, B, Care internally tangent to (P), (Q) at PA, QA, PB, QB, PC, QC, respectively. A1, A2 are the

intersections of B and C, and similarly define B1, B2, C1, C2. Denote by A the circumcircle of

PAQAA1A2 (it is easy to see its cyclic), and similarly B, C.

a) Show that A, B, Care coaxial, their centers lying on a line .

b) Show that the envelope of as A varies is a conic.

Problem 14: Given B, Cmixtilinear excircles ofABCtouch the lines BC,AC,AB at D, G; E;

F resp. Points M, Nare midpoints ofGF,DE. M E cuts N F at P. The line GFcuts the line DE

at T, the lines T A, P A cuts the line BCat H, Q resp. Prove that BQ = C H.

Problem 15: Given M , Nare the C, B excenters ofABC, the A mixtilinear excircles ofABC

touch the lines AB, ACat H, Iresp. Prove that M I , H N , B C are concurrent.

Problem 16: Given DEF

is the orthic triangle of ABC

. Ia

, Ib, Ic are the

A, B , C excentersofAEF, BDF, CDE. IaIb cuts DEat M. IaIc cuts DF at K. IcIb cuts F Eat L. Prove that

L, F, K are collinear.

Problem 17: (O1), (O2), (O3) are the A, B, C mixtilinear incircles ofABC. They touch (ABC)

at D , E , F reps. F O1 cuts DO3 at Z. F O2 cuts EO3 at X. DO2 cuts EO1 at Y. Prove that

Y O3, XO1, ZO2 are concurrent.

Problem 18: The B, Cmixtilnear excircles ofABC touch CB at D, W resp. They meet the

extentions ofAC, ABat G, Hresp. The incircles ofABCtouchesAB, ACat Q, N resp. W H cutsDG at M. QD cuts W Sat P. Prove that BAM= P AC.

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Problem 19: Pis arbitrary point on the side AB of aABC with circumcircle(O)and C P cuts

(O) again at D..(I1), (I2) are ordinary incircles ofAPC, BP C with incenters I1, I2.. (J1), (J2)

are Thebault circles with centers J1, J2, tangent to the rays P D, P A resp P D, P B and internally

tangent to the circumcircle (O). Prove that the center lines I1I2, J1J2 intersect on the line AB.

Problem 20: LetABCbe a triangle with circumcircle(O).P, Qare two point on BC.AP, AQcut

(O) again at M , N, respectively.(I1), (I2), (I3), (I4)are incircles of triangle P AB, QAC,QAB, P AC,

respectively.(J1), (J2), (J3), (J4) are Thebault circles tangent to the rays P B, P M; QC, QN; QB,

QN;P C,P M, respectively and internally tangent to the circumcircle (O). Prove thatI1I2,I3I4,J1J2,

J3J4 and BCare concurrent.

Problem 21: Let ABCbe a triangle and a point P. P A, P B , P C cut BC,CA,AB at A1, B1, C1,

respectively. The intersections of B1C1 and BC, C1A1 and CA, A1B1 and AB are collinear on a

line d. P A, P B , P C cut d at A2, B2, C2, respectively. is a line cutting BC,CA,AB at A3, B3, C3,

respectively. B3B2 C3C2 A4, C3C2 A3A2 B4, A3A2 B3B2 C4. B4C4, C4A4, A4B4 cutd at A2, B2, C2 ,respectively. Let A5, B5, C5 be the points such that cross ratio (B4C4A2A5) =

(C4A4B2B5) = (A4B4C2C5) =1.

a) Prove that A4A5, B4B5, C4C5 are concurrent at point Q.

b) P Q cut , d at R, S, respectively. Prove that (PQRS) =1.

Problem 22: Let ABCbe a triangle with orthocenter H. M, N are points on CA,AB, respec-

tively. Constructed triangle P BC HN Mand they are in opposite directions such that P and

A are in the same side with BC. Prove that P HM N.

Problem 23: Let ABCbe a triangle and a point P. Q is isogonal conjugate ofPwith respect to

triangle ABC. Kis midpoint ofP Q. H is projection ofP on BC. Ray HPcuts circle(Q, 2KH) at

A. Circumcircle(HQA) (K,KH) ={H, M}. Prove that HKpasses through M.

Problem 24: Let ABCbe a triangle and A-excircle touches BCat D. d is a line passing through

D.d cut CA,ABatE , F, respectively.M isE-excenter of triangleDCE,N isF-excenter of triangle

DBF, P is incenter of triangle AEF. Prove that A, M , N , P are concyclic.

Problem 25: A circle (O) through B,C is tangent to the incircle (I) ofABC at X. (O) cut

CA,AB at P,Q and BP meet CQ at N. AHis the altitude ofABC. M is the midpoint ofAH.

I1, I2 are the incenters ofP N Q,CN B. AX meet (O) at Y (different from X).Prove that:

(1) D,I2,M,Xare collinear, where its well known that D,M,Xare collinear.

(2) I,I2,Yare collinear.

(3) X,I1,I2,Y are cyclic.

(4) when a tangent to (I) at X cut BC at K, a circle (K,KX) pass I1.

Problem 26: Let ABCbe a triangle with circumcenter O. P, Q are two isogonal conjugate withrespect to triangle such that P, Q, O are collinear. Prove that four nine-point circles of triangles

ABC,APQ,BPQ,CPQ have a same point.

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Problem 27: Let ABC be a triangle and O is circumcenter I is incenter. OI cuts BC,CA,AB

at D, E , F . The lines passing through D, E , F and perpendicular to BC,CA,AB bound a triangle

M N P. Let F e , G be Feuerbach points of triangle ABC and M N P. Prove that OIpasses through

midpoint ofF eG.

Problem 28: Let(O1)and (O2)be two circles andd is their radical axis. Iis a point on d.IA, IB

tangent to(O1), (O2)(A (O1), B (O2)) andA, B have same side withO1O2, respectively.IA, IB

cut O1O2 at C, D. P is a point on d. P C cut (O1) at M, N such that N is between M and C. P D

cut (O2) at K, L such that L is between K and D. M O1 cuts KO2 at T. Prove that T M=T K.

Problem 29: Let ABCbe a triangle with circumcircle (O) and P, Q are two isogonal conjugate

points. A1, B1, C1are midponts ofBC, CA, AB, resp.P A, P B , P C cut(O)again at A2, B2, C2, resp.

A2A1, B2B1, C2C2 cut (O)again at A3, B3, C3. A3Q, B3Q, C3QcutBC,CA, ABat A4, B4, C4. Prove

that AA4, BB4, CC4 are concurrent.

Problem 30: Let ABC and ABC be two triangles inscribed circle (O). Prove that orthopoles

of BC, CA, AB with respect to triangle ABC and orthopoles of BC,CA,AB with respect to

triangle ABC lie on a circle.

Problem 31: Let ABCbe a triangle inscribed circle (O)and orthocenter H. Pis a point on (O).

dis Steiner line ofP. M is Miquel point ofd with respect to triangle ABC. Prove that M , P, H are

collinear iffOP d.

Problem 32: LetABCbe a triangle with orthocenter Hand circumcenter O.()is a circle center

O and radius k.(K), (L) are two circle passing through O, Hand touch () at M, N.

a) Prove that (K), (L) have the same radius.

b) Prove that M Npasses through H.

c) Prove that circle()center Hradius kalso touches(K)and(L)at P, Q and P Q passes through O.

Problem 33: Let ABC be triangle inscribed (O). (Oa) is A-mixtilinear incircle of ABC. Circle

(a) other than (

O) passing through

B, Cand touches (

Oa) at

A

. Similarly we haveB, C

. Provethat AA, BB , CC are concurrent.

Problem 34: Let O be circumcenter of triangle ABC. D is a point on B C.(K) is circumcircle of

triangle ABD. (K) cuts OA again at E.

a) Prove that B , K , O, E are concyclic on circle (L).

b) (L)cuts AB again at F. G is on (K) such that E G OF. GK cuts ADat S. SO cuts BCat T.

Prove that O, E , T , C are concyclic.

Problem 35: (O1), (O2), (O3) are the A, B, C mixtilinear incircles ofABC. They touch (ABC)at D, E , F resp. (O2), (O3) touch BC at W, V resp. Iis the incenter ofABC. DV cuts F I at Y,

EF cuts BCat Z, DW cuts EI at X. Prove that X, Y , Z are collinear.

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Problem 36: The A mixtilinear incircles ofABC touches AB, AC, (ABC) at G, H, D resp. BI

cuts DG at S. DHcuts ICat T. Prove that ST GH.

Problem 37: The A, B , C mixtilinear incircles of ABC touch the ABC at D , E , F resp and

touchBC,CA, ABat W, V , H, T , U , Gresp. F W cutsDGat N, V E cutsDHat Y , C Y cutsF T at

K, BN cuts EU at P. Prove that P KB C.

Problem 38: ABC, AB > BC > CA.DEF is the orthic triangle ofABC. Ia, Ib, Ic are the

A , B , C excenters of triangles AEF, BDF, CDEresp. Prove that:

a) BIaC+ AIbC BIcA= 180.

b) BIa

CIa.

CIb

AIb.

AIc

BIc.

Problem 39: AB,BE,CFare concurrent. The angle bisector of AEB cuts the angle bisector ofADB at Z. The angle bisector of

BF Ccuts the angle bisector of

BE C at X.The angle bisector of

ADCcuts the angle bisector of AF Cat Y. Prove that AX,BY, CZare concurrent.

Problem 40: AB,BE,CFare concurrent. The angle bisector of AEB cuts the angle bisector ofAF C at T. The angle bisector of BF Ccuts the angle bisector of BDA at K.The angle bisector ofCEB cuts the angle bisector of ADC at J. Prove that K E , F J, DTare concurrent.

Problem 41: (O1), (O2), (O3) are the three mixtilinear incircles ofABC. They cut (ABC) at

D , E , F resp. Their points tangency on BC,CA,AB are W, V , H, T , Y , G resp. U T cuts EF at X,

DF cuts GW at Y. Prove that AXB Y.

Problem 42:DEFis the orthic triangle ofABC.Ia, Ib, Icare theA, B , C excenters ofAEF,

BDF, CDE. Prove that DIa, EIb, F Ic are concurrent.

Problem 43: (O1), (O2) are the B, C excircles ofABCresp, the points of tangency are D, K,

E, F, L, G. GK, DLcuts (O1), (O2) at M, N resp. M L cuts N Kat P. Prove that AP B C.

Problem 44: Circle (O) and circle (P) are externally tangent at T. A, B , M , N are on (O) suchthat AN and BMare tangent to (P)at F, E resp. N T , M T meet(P) at R, Qresp. F R cuts E Qat

U. Prove that U T R= U T Q

Problem 45:(O1), (O2)are the B, Cmixtilinear excircles ofABCresp. The points of tangency

are G, F, E , D. M is the midpoint ofDE. M F cuts O1O2 at W. Prove that BW O1O2.

Problem 46: The Mixtilinear excircles ofABCcuts each side at H , I , D , E , F , G. IE cuts DF

at K. HF cuts GEat J. JC cuts BKat S. Prove that IJ,HK,ASare concurrent.

Problem 47: (S), (P), (Q) are the A , B, C mixtilinear incircles ofABC resp. A mixtillinear

touches the circumcircle of ABC at D. DQ cuts (Q) at M. DP cuts (P) at N. Prove that

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M NP Q.

Problem 48:(O1), (O2), (O3)are the mixtilinear excircle ofABC,H, I , D , E , F , G are the points

of tangency onBC, CA,ABresp.GO2cutsDO3at X, defineY , Zsimilarly. Prove that AX,BC,CZ

are concurrent.

Problem 49: The three excircles ofABCtouch three sides at I, J, D, G, H , E resp. M , N , F , T , P , K

are the midpoints ofIJ, HG, DJ, GD,HE, EL. Prove that P F , K T , M N are concurrent.

Problem 50: DEF is the orthic triangle ofABC. The incenters ofADF, BDE, CEF

are K, I , X resp. The points tangency are G, M, W, Q, N, H . I M cuts KN at Y, K H cuts IGat Z.

Prove that X, Y , Z are collinear.

Problem 51: The B , Cexcircles ofABCtouchAC, ABat F, Gresp. They touch B Cat Dand

E resp. EF,DG cut the two circles at L, M. C G, F B cut the two excircles at T, S. T B cuts SC atH. M B cuts LC at P. Prove that H, P, A are collinear.

Problem 52: The A mixtilinear incircle ofABCtouchesAB,ACatG, Hresp.B , Cmixtilinear

incircles touchBCatV, W resp.M, Nare the midpoints ofGW, HV resp.GN cutsH MatP. Prove

that MV, WN, A Pare concurrent.

Problem 53: The mixtilinear incircles wrt B, C for ABCtouches the(ABC) at E , Fresp. The

mixtilinear incircle wrt A touches AB, AC at G, H resp. F H cuts GEat P. T is on BCsuch that

AB+ BT =p. Prove that BAT = P AC.

Problem 54: The mixtilinear excircles of ABC touch three sides at H, I, D, E, F, G resp.

J, L, N, K , M, Pare midpoints ofBC,CA,AB,EF,GH, IDresp. Prove that M L, P N , J K are con-

current.

Problem 55: The mitilinear incircle and mixtilinear excircle wrt A ofABC touches the ABC

at T, X resp. AXcuts the mixtilinear incircle wrt A at S. The incircle ofABC touches AB, AC

at E, Fresp. Prove that

ES

SF =

T F

T E.

Problem 56:DEFis the orthic triangle ofABC. The incircles of trianglesBDE,CEF, ADF

are (I), (J), (K) resp. (I), (K) cut AB,BC, CAat H , G, L, S resp. KS cuts J H at M. J G cuts IL

at N. Prove that MN, AB,LKare concurrent.

Problem 57: A, B , C, D are on (O). (P) externally touches (O) at G, it also touches AB,BC at

F, Eresp. The angle bisectors of FDC, DC Ecut EF at L, Kresp. Prove that ABG = LKG

Problem 58: B, Care on (O). The A mixtilinear incircle ofABC touches (O) and AB at P, D

resp. (BDP) cuts BC, CP at H, G resp. Prove that BG DH.

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Problem 59: The A mixtilinear incircle ofABC touches AB,AC at G, Hresp. It also touches

the (ABC) at D.(BGD) and (HCD) cut BCat J, Iresp. Prove that IGJ= IHJ.

Problem 60: Let O be the circumcenter of the acute triangle ABC. Suppose points A, B and

C are on sides BC, CAand AB such that circumcircles of triangles ABC, BCA and CAB pass

through O. Let la be the radical axis of the circle with center B and radius BCand circle with

centerC and radiusCB. Define lband lc similarly. Prove that lines la, lbandlc form a triangle such

that its orthocenter coincides with orthocenter of triangle ABC.

Problem 61: Let ABCbe a triangle. P is a point inside ABC. AP, BP, CP cut BC,AC,AB at

E , F , D respectively. The angle bisector ofADCcuts the angle bisector ofAF B at K. The angle

bisector ofBDCcuts the angle bisector ofAEB atI. The angle bisector ofAECcuts the angle

bisector ofCF B at J.Prove that CJ, BI, AKare concurrent.

Problem 62: Let AA

, BB

, CC

be three diameters of the circumcircle of an acute triangle AB C.Let Pbe an arbitrary point in the interior ofABC, and let D, E , F be the orthogonal projection

ofP on BC,CA, AB, respectively. Let Xbe the point such that D is the midpoint ofAX, let Y be

the point such that E is the midpoint ofBY, and similarly let Zbe the point such that F is the

midpoint ofCZ. Prove that triangle XY Zis similar to triangle ABC.

Problem 63: Let, 1, 2be three mutually tangent circles such that 1, 2are externally tangent

at P, 1, are internally tangent at A, and , 2 are internally tangent at B. Let O, O1, O2 be the

centers of , 1, 2, respectively. Given that Xis the foot of the perpendicular from P to AB , prove

that O1XP = O2XP.

Problem 64: LetABCbe a triangle with altitudesAD,BE, CF, mediansAM,BN, CPand three

Feuerbach points Fa, Fb, Fc, (E) is nine points circles. Tangents at Fa, Fb, Fc of(E) intersect base a

triangle XY Z.

a) Prove that two triangle XY Zand DEFare perspective.

b) Prove that two triangle XY Zand M N Pare perspective.

Problem 65: Suppose that the B, C

mixtilinear incircles (with centers atOb, O

c ,respectively) ofthe triangle touch the Circumcircle of triangle AB Cat D, E . respectively. The lines E Ob, DOcmeet

at F. Show that the line AFpasses through the incenter of the triangle.

Problem 66: Let ABCbe a triangle with circumcircle (O) and a point P. P A, P B , P C cut (O)

again at D, E , F . Let da, db, dc be Simson line ofD, E , F with respect to triangle AB C, respectively.

Prove that da, db, dc intersect base a triangle that is orthologic with triangle ABC.

Problem 67: Let ABCbe a triangle with mixtilinear incircles (Oa), (Ob), (Oc). (Oa) cuts BC at

A1, A2 such that A1 is between B, A2. (Ob) cuts CA at B1, B2 such that B1 is between C, B2. (Oc)cuts AB at C1, C2 such that C1 is between A, C2. Prove that A2B1, B2C1, C2A1 intersect base a

triangle that is perspective with triangle ABC.

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Problem 68: Let ABCbe a triangle and a point P. (P) is a circle center (P). Let D, E , F be

inverse points ofA, B , C with respect to(P).X, Y , Z lie onBC,CA, ABsuch thatDXPA , E Y

P B , F Z P C. Prove that X, Y , Z are collinear.

Problem 69: Let DEFbe pedal triangle of a point P with respect to triangle ABC. X, Y , Z lie

on E F, FD, DE , respectively such that P X P A, P Y P B , P Z P C. Prove that X, Y , Z are

collinear.

Problem 70: Let Kbe orthopole of a line d with respect to triangle ABC. x, y are two perpen-

dicular lines passing throughK. x, y cut B CatA1, A2,x, y cutC Aat B1, B2,x, y cutAB at C1, C2,

x, y cut d at X, Y. Assume that midpoints ofA1A2, B1B2, C1C2 are collinear on a line then that line

passes though midpoint ofXY.

Problem 71: In the acute-angled triangle AB Cwith orthocenter Hand circumcircle k is drawn alinel throughH(non-intersectingAB).KandL are the intersection points ofk and l (Kis from the

smaller arcAC,L is from the smaller arc B C). M, NandPare the feets of the perpendiculars from

the verticesA,B andCtol, respectively (M,N,Pare internal tok). Prove thatP H=|KMLN|.

Problem 72: Let(I)be a circle inside circle (O). P, Qare two points on(I). There are two circles

(K) and (L) which is passing through P, Q and intouch (O) at M, N, respectively. Prove that M N

always passes through a fixed point when P, Q move on (I).

Problem 73: Let AB Cbe a triangle with altitudes AD,BE, CF.(N)is nine point circle. X, Y , Z

are in inverse points ofA, B , C with respect to (N). Prove that DX, E Y, FZare concurrent.

Problem 74: Let ABCbe triangle with incircle (I). (I) touches BC,CA, AB at D, E , F , respec-

tively. (1) and (2) are two circles center I. Let M , N , P are in inverse points of D , E , F with

respecto circle(1). Let X, Y , Z are in inverse points ofA, B , C with respecto circle (2). Prove that

M X , N Y , P Z are concurrent.

Problem 75: LetABC

be a triangle and a pointP

.DEF

is pedal triangle ofP

with respect toABC. (P) is circle center P. Let X, Y , Z be polars of lines E F, FD, DE with respect to circle (P),

respectively. Prove that AX, BY,CZare concurrent.

Problem 76: Given triangle AB Cand its circumcircle (O).(O1)is a circle that tangents AB, AC

at N, Kand circle(O).Circles(O2) and(O3) are defined similarly. According to the given diagram,

prove that BN M LKC.

Problem 77: From the three sides ofABCdraw rectangles ACDE,BCIH,ABGF.V, W, Y are

the midpoints ofH E, F I , DG. Prove that the lines perpendicular ro EF,GH,DIthat go throughY , W, Vresp. are concurrent.

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Problem78: Let ABCDEFbe bicentric hexagon with incircle (I) and circumcircle (O).

a) Prove that AD,BE, CFare concurrent at point Kon OI.

b) Constructed circles(K1), (K2), (K3), (K4), (K5), (K6)as in figure. Let G, H , J, L, M, Nbe incenters

of triangles KAB,KBC,KCD,KDE,KEF,KFA, respectively. Prove that X , G, L, T; Y , H, M , U ;

Z, J, N, Vare concyclic on circles (O1), (O2), (O3), respectively.

c) Prove that three circles (O1), (O2), (O3) are coaxal with radical axis is OI.

Problem 79: Let ABCD be circumscribed quadrilateral and M is its Miquel point. (M) is a cir-

cle center M. Let A, B, C, D invert A, B , C, D through (M), respectively. Prove that ABCD is

circumscribed quadrilateral.

Problem 80: Let ABCbe a triangle with circumcenter O. d is a line passing though O. d cuts

circumcircle of triangle OBC,OCA,OAB again at X, Y , Z , respectively. A circle center O cuts rays

OA,OB,OCat A, B, C, respectively. Prove that AX, BY, CZare concurrent.

Problem 81:ABC,ADthe angle bisector.Mthe midpoint ofAD.O1, whose diameter is AB ,

intersectsC MatF.O2, whose diameter isAC, intersectsBMatE. Prove that:M E F Dconcyclic

and BCEF concyclic.

Problem 82: Let Band Dbe points on segments[AE]and[AF]respectively. Excircles of triangles

ABF andADEtouching sidesB FandDEis the same, and its center is I.BF andDEintersects at

C. Let P1, P2, P3, P4, Q1, Q2, Q3, Q4 be the circumcenters of triangles IAB, IBC, ICD, IDA, IAE,

IEC, ICF, IF A respectively.

a) Show that points P1, P2, P3, P4 concylic and points Q1, Q2, Q3, Q4 concylic.

b)Denote centers of theese circles as O1 and O2. Prove that O1, O2 and Iare collinear.

Problem 83: ABCis a non-isosceles triangle. TA is the tangency point of incircle ofABC in the

side BC (define TB,TC analogously). IA is the ex-center relative to the side BC (define IB,IC anal-

ogously). XA is the mid-point ofIBIC (define XB,XCanalogously). Prove that XATA,XBTB,XCTC

meet in a common point, colinear with the incenter and circumcenter ofABC.

Problem 84: Line intersects sides (or extension) of triangleABC

(or extension) withA1

,B1

,C1

.Let O,H,Oa,Ha,Ob,Hb,Oc,Hc be the circumcenters and orthocenters of triangles ABC , AC1B1 ,

BA1C1 , CA1B1 respectively. Prove that midperpendicular ofOH,OaHa,ObHb, OcHc intersects one

point.

Problem 85: Given (O1), (O2) are B excircle and C excircle ofABC, they touch AB,AC at

R , T , E , F , the line CR and BT intersect (O1), (O2) at H, P resp, the line CR intersects BT at J,

the line HR intersects EC at L, the line P T intersects BF at K, the line P L intersects KH at I.

Prove that BF, IJ, CEare concurrent.

Problem 87: Let ABCDE be bicentric pentagon with incircle (I) and circumcircle (O). M, N,

P, Q, R are intersections of diagonals a figure. Constructed circles(K1), (K2), (K3), (K4), (K5) as in

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figure. Prove that X M , Y N , Z P, T Q, U R are concurrent at a point Son OI.

Problem 88: Let ABCbe a triangle and P, Q are two arbitrary point. A1B1C1 is pedal triangle

ofP with respect to triangle ABC. A2, B2, C2 are symmetric ofQ through A1, B1, C1, respectively.

A3, B3, C3are reflection ofA2, B2, C2throughBC,CA, AB,respectively. Prove that Q, A3, B3, C3are

concyclic.

Problem 89: Let ABCDbe a parallelogram.(O)is circumcircle of triangle AB C. P is a point on

BC. Kis circumcenter of triangle P AB. L is in AB such that KL B C. C Lcuts(O)again at M.

Prove that M , P , C, Dare concyclic.

Problem 90: Let ABC be triangle and P, Q are two isogonal conjugate points with respect to

triangle ABC. Prove that circumcenter of the triangles P AB, P AC,QAB, QACare concyclic.

Problem 91: Let O be circumcenter of triangle ABC. D is a point on B C.(K) is circumcircle oftriangle ABD. (K) cuts OA again at E.

a) Prove that B , K , O, E are concyclic on circle (L).

b) (L)cuts AB again at F. G is on (K) such that E G OF. GK cuts ADat S. SO cuts BCat T.

Prove that O, E , T , C are concyclic.

Problem 92: LetABCDbe a cyclic quadrilateral. Circle pass through A, DcutsAC, DBat E , F.

Glies on ACsuch that BG DE, H lies on BD such that CHAF. AF DEX,DE CH

Y , C H GB Z,GB AF T. M lies on AC. N lies on BD such that M NAB , P lies on AC

such that N P BC, Q lies on BD such that P Q CD. dM passes though M, dP passes though

P such that dM dP DE. dQ passes though Q, dN passes though N such that dQ dN AF.

dQ dMU, dM dNV , dN dP W, dP dQS. Prove that X U, ZW, S Y, TVare concurrent.

Problem 93: Let ABCbe a triangle with circumcircle (O). D, Eare on (O). DE cuts BCat T.

Line passes though Tand parallel to AD cuts AB,ACat M, N. Line passes though Tand parallel

to AE cuts AB, AC at P, Q. Perpendicular bisector of M N, P Q cut perpendicular bisector ofBC

at X, Y, resp. Prove that O is midpoint ofXY.

Problem 94: Let ABCbe triangle with circumcircle (O, R). P, P are two isogonal conjugate

points with respect to triangle ABC. Q is reflection ofP through BC. AP, AP cut (O) again at

D, D. DQ cuts (O) again at E. EP cuts (O) again at E. Prove that AED E.

Problem 95: Let ABCbe a triangle with circumcircle (O). Pis a point on line BC outside (O).

Tis a point on APsuch that B T , C T cuts(O) again at M , N, resp, then M NP A. Q is reflection

ofP through M B, R is reflection ofP through N C. Prove that QR B C.

Problem 96: Let AB Cbe a triangle and a circle ()passes through B , C. The circle(K)touchesto segment AC, AB at E, Fand externally tangent to () at T. Prove that intersection other than

Tof circumcircle (T EC) and (T F B) is incenter of triangle ABC.

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Problem 97: Let ABCbe triangle with incenter I. A circle ()passes throughB, C. T is a point

on (). Circumcircle (BI T) cuts AB again at F. Circumcircle (CIT) cuts ACagain at E. (K) is

cirumcircle of triangle T EF.

a) Prove that K lies on AI.

b) (K) cuts AB,ACagain at P, Q, resp. Prove that PQ, EF, AIare concurrent.

Problem 98: Let ABCbe triangle and a point P. A1B1C1 is pedal triangle ofPwith respect to

triangle ABC. A2B2C2 is circumcevian triangle ofP. A3B3C3 is pedal triangle ofPwith respective

to triangle A2B2C2. Prove that A1B1C1 and A3B3C3 are persective if only ifA1B1C1 and A2B2C2

are perspective.

Problem 99: Let AB Cbe triangle with circumcircle (O)and a pointD.(O1), (O2)are circumcir-

cles of triangles ABD, ACD, resp. DO1 cuts (O2) again at E. DO2 cuts(O1) again at F.

a) Prove that A, E , F, O1, O2 lie on a circle (K).b) DB cuts (O2) again at M, DC cuts (O1) again at N, EM cuts (K) again at P, F N cuts (K)

again at Q. Prove that B , P, F ; C, Q, E ; P , O, O1; Q, O, O2 are collinear, resp.

Problem 100: LetAB Cbe triangle with circumcircle(O).Pis a point andP A, P B , P C cuts(O)

again at A.B, C. Tangent at A of(O) cuts BC at T. T P cuts (O) at M, N. Prove that triangle

ABC and AM Nhave the same A-symmedian.

Problem 101: LetAB Cbe triangle. A circle(K)passing throughB , CcutsCA,ABatE , F.B E

cuts CF at G. AG cuts BC at H. L is projection ofH on EF. M is midpoint of BC. M K cuts

circumcircle (KEF) again at N. Prove that LAB = N AC.

Problem 102: Given AMB, AKC, BN C are isosceles right triangles. Their incenters are

S , T , R resp. M = N = K = 90. U, X , J are the midpoints of S R , S T, TR resp. Prove the lines

through X, U, Jthat are perpendicular to BC,CA,AB resp are concurrent.

Problem 103: Let M, Nbe two points interior to the circle (O) such that O is the midpoint of

M N. Let

S be an arbitrary points lies on (

O) and

E, F are the second intersections of the lines

SM,SN with (O), resp. The tangents in E, Fwith respect to the circle (O) intersect each other at

I. Prove that the perpendicular bisector of segment M Npasses through the midpoint ofSI.

Problem 104: The A mixtilinear incircle and the A mixtilinear excircle ofABCtouch the(ABC)

at D, Q resp. QD cuts BCat P. Prove that AP AO.

Problem 105: In an acute triangle ABC, M is the midpoint of BC, and P is the point on BC

such that BAP = CAM. The tangent of A to the circumcircle of ABC meets BC at N. The

circumcircles ofN AB and CAM meet again at Q. The line through Q perpendicular to P Q meetsAB and AC at D and E, respectively, and BE meets CD at R. QR meets AN and AM at F and

G, respectively, and DG meets F Eat S. Prove that ASB C.

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Problem 106: I is the reflection of incenter I over BC, EH BC,DS BC,DG AB,

EFAB . HF cuts GSat M. Prove that A, M , I are collinear.

Problem 107: The three mixtilinear excircles of ABC touch the three sides BC,CA,AB at

H , I , D , E , F , G resp. IEcuts DFat K, HF cuts GEat J, J Bcuts KCat Q. Prove that HF,EI,AQ

are concurrent.

Problem 108:(Oa), (Ob), (Oc)are theA, B , C mixtilinear incircles ofABCresp. They touch(O)

at D, E , F resp. Let Xbe the intersection ofF Ob, EOc. Similarly we have Y, Z. Prove that DEF

and XY Zare perspective.

Problem 109:(O1), (O2are theB, Cmixtilinear incircles ofABC. They touchBCatG, Dresp.

(O1) touches AB at F. (O2) touches ACat E. DE, GF cut O1O2 at K, J resp. CK cuts BJ at N.

Prove that AN bisectsBAC.

Problem 110: The incircle ofABC touches BC,CA,AB at D, E , F resp. AD cuts EF at G.

BEcuts DF at L. GL cuts AC at M. DEcuts CF at K. AD cuts ZF at J. KM cuts EF at H.

Prove that GL, HJ, BCare concurrent.

Problem 111:(O1), (O2)are theB, Cmixtilinear excircles ofABC. They touchAB, ACatF, E

resp. S, Hare resp on AB,ACsuch that O1HAC, O2SAB . O1H cuts O2Sat P. Prove that:

a) E F, O1O2, SHare concurrent.

b) AEF = ASH.

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111 Nice Geometry Problems