Tran Quang Hung - Red Geometry

8/17/2019 Tran Quang Hung - Red Geometry

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Tran Quang Hung - Red geometry 1

Red geometry by Tran Quang HungProblem 1. Let P is a point on A−bisector of triangle ABC , M , N is projection of P on AB, AC such that BM . CN = P H . P A when H is intersection of M N with P A. K is point on BC such that ∠M KN = 90◦ prove that K , P , O are collinear when O is circumcenter of ABC .

Problem 2. Let ABC be a triangle and P is an arbitrary point, ABC is pedal triangle of P , through A, B, C draw d a P A, d b P B, d c P C , H is orthocenter of triangle ABC . d a ∩ HA = A

,d b ∩HB = B

, d c ∩HC = C , circumcircle of ABC intersects nine −point circle of triangle ABC

at F , F prove that the circles with diameters ( AA), ( BB), ( C C ) are through F or F .

Problem 3. Let ABC be a triangle and ABC is pedal triangle of an arbitrary point P , let P B,P C intersect C A, AB ay Y , Z , resp. BY , CZ intersect CA, AB at E , F , resp. P B intersects EF at D. Prove that A, B, D are collinear.

Problem 4. Let AH be altitude of triangle ABC , d is a line which is through A, d cut circumcircle of ABC second point D, M , N are projections of B, C on AD, resp, P is projection of D on BC .

a) Prove that M , N , H , P are concyclic.b) Let O be circumcenter of AB C , D P ∩AO = O , Y , Z are projections of O on AB , AC . Prove

that Z Y H P is isoceles trapezoid.

Problem 5. Let P is a point on altitude of triangle ABC and B, C are projections of P on AC ,AB.

a) Prove that B, C , B, C are concyclic with center I .b) Let AI ∩ BC = J and K is projection of J on line BC , Q is a point on line AJ . M , N are

midpoint of BC and CB. The perpendicular to line QB, QC through M , N , resp intersect at point H . Prove that HB = H C .

Problem 6.

Let ABCD

be a cyclic quadrilateral M

, N

, P

, Q

are midpoins of AB

, BC

, CD

, DA

a) Prove that the lines is through M , N , P , Q which are perpendicualr to CD, DA, AB, BC concurrent at point H . It called be orthocenter of ABCD.

b) Let d and d’ be two perpendicular lines through H . d cuts AB, BC , CD, DA, AC , BD at X ,Y , Z , T , U , V and d’ cuts AB, BC , CD, DA, AC , BD at X , Y , Z , T , U , V . Let E , F , K , L,I , J be midpoints of XX , Y Y , ZZ , T T , U U , V V . Prove that EK , F L, IJ are concurrent.

Problem 7. Let ABC be a triangle and take three similar rectangles inscribed ABC (a rectangle inscribed ABC has two vertices on side a and two remaining vertices on two other sides of ABC ).Let A, B, C be center of these rectangles (cyclic with A, B, C ), respectively.

a) Prove that AA, BB , CC are concurrent.

b) Let A

, B

, C

are reflections of A

, B

, C

through BC , CA, AB, resp. Prove that AA

,BB , CC are concurrent.

Problem 8. Let ABCD be convex quadrilateral inscribed a Ellipse with center O. M , N are midpoint of AC , BD. Through A, C draw two line a, c parallel to ON , Through B, D draw two line b, dparallel to OM . Let a intersect b, d at Q, R, resp, c intersect b, d at P , S , resp. AC ∩BD = I .Let X , Y , Z , T be centroids of triangle IAB, IBC , ICD, IDA. Prove that four lines XQ, ZS ,Y P , T R are concurrent.


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Problem 9. Let (O1), (O2) be two orthogonal circles, and P is a point on (O2), draw two tranversal P M N and P EF ( M , N , E , F ∈ (O1), this means P , M , N and P , E , F are collinear). Let M E ∩ N F = K , P O1 ∩ (O2) = L then P L ⊥ LK .

Problem 10. Three medians AA, BB , CC of triangle ABC are concurrent at G. Take A1,

A2 ∈ AA such that

−−−→A1A2 =

1

2

−−→AA similarly we have B1, B2, C 1, C 2. Draw two lines through A1,

A2 and perpendicular to AA, similarly for cyclically B, C , we get six lines. Prove that six lines intersect, respectively (cyclically A, B, C ), base a cyclic hexagon.

Problem 11. Let ABC be a triangle and A BC be a Cevian triangle of point P . P A, B C intersect circumcircle ( ABC ) at {A, A1} and {A

, A2}, resp. A1A2 intersect BC at H . Tangent of

( ABC ) at A2 intersects BC , AB, AC , AH at I , J , K , L, resp. IA intersect A1A2, A2B

, A2C

at I , M , N , resp. Prove that N J , KL, MI are concurrent.

Problem 12. Let H be orthocenter of triangle ABC and A1B1C 1 is pedal triangle of H wrt ABC ,A2B2C 2 is pedal triangle of H wrt A1B1C 1. Take A3, B3, C 3 on ray HA2, HB2, HC 2 such that HA3 = H B3 = H C 3.

a) Prove that radical center of circles ( A1, A1A3), ( B1, B1B3), ( C 1, C 1C 3) lie on Euler line of triangle ABC .

b) Prove that radical center of circles ( A, AA3), ( B, BB3), ( C , CC 3) lie on Euler line of triangle ABC .

Problem 13. Let ABCDEF be a hexagon inscribed circle (O) such that AD, BF , CE are concur-rent. Let P is an arbitrary point, through A, B, C , D, E , F draw six lines perpendicular to P A,P B, P C , P D, P E , P F , resp, they intersect (O) at second point A , B, C , D, E Prove that AD,BF , C E are concurrent.

Problem 14. Let ABCDEF be a cyclic hexagon and point P . Let A, B, C , D, E , F be

circumcenter of triangles P AB, P BC , P CD, P DE , P EF , P F A. Prove that A

D

, B

E

, C

F

are concurrent.

Problem 15. Let ABC be a triangle and ABC be Cevian triangle of point O. M , N , P are midpoint of AB, BC , OA. d is a line through B. M B, M P , MN intersect d at E , F , G, resp.BC ∩ BC = {I }. IF ∩ GA = {J }. Prove that E , J , C are collinear.

Problem 16. Let ABC be a triangle and point P a) Through midpoint of BC , CA, AB draw lines parallel to P A, P B, P C prove that those lines

are concurrent at point O.b) Let ABC be cevian triangle of P . Through midpoint of P A draw line parallell to BC , it cuts

BC at Q. OB ∩

AC = {

B1}

, OC ∩

AB = {

C 1}

. Prove that P Q

B1C 1

Problem 17. Let ABC be a triangle and point P , A1B1C 1 is cevian triangle of P . A2, B2, C 2 are midpoints of BC , CA, AB. Q is an arbitrary point. QA2, QB2, QC 2 intersect P A, P B, P C at A3,B3, C 3, resp. Let d a, d b, d c be the lines through A, B, C and parallel to QA2, QB2, QC 2, resp. d a,d b, d c intersect BC , CA, AB at A4, B4, C 4. Let A5, B5, C 5 be the point such that A4, B4, C 4 be midpoints of AA5, BB5, CC 5, reps. Let Ga, Gb, Gc be centroid of triangle A3BC , B3CA, C 3AB,assume that AGa, BGb, CGc are concurrent, prove that A5A1, B5B1, C 5C 1 are concurrent.


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Problem 18. Let ABC be a pedal triangle of arbitrary point P with respect to triangle ABC . P ∗

is isogonal conjugate of P with respect to triangle ABC . R p is circumradii of triangle ABC . The

rays [ BP , [ C P intersect circle ( P ∗, 2R p) at B, C , respectively. Prove that BB , CC intersect

on circle ( P ∗, 2R p).

Problem 19. Given triangle ABC and A1B1C 1 is cevian triangle of an arbitrary point P . P ∗ is

isogonal conjugate of P . A2, B2, C 2 are reflections of P through B1C 2, C 1A1, A1B1, respectively.AA2, BB2, CC 2 are concurrent at BP . Prove that circumcenter of triangle A2B2C 2 lies on line P ∗BP .

Problem 20. Let ABC be a triangle with its circumcircle (O). Let P and Q be arbitrary points such that P , O, Q are collinear. A1B1C 1 is the pedal triangle of P wrt ∆ABC , A2B2C 2 is the circumcevian triangle of Q wrt ∆ABC . Show that (P A1A2), (P B1B2), (P C 1C 2) are coaxal.

Problem 21. Let ABCD be a quadrilateral, AC ∩ BC = {O}. d, d’ are the lines connecting A, C .P ∈ d, Q ∈ d’ such that P , O, Q are collinear. DP ∩ d’ = {M }. CQ∩ d = {N } prove that M N is always through a fix point when P , Q move.

Problem 22. Let ABC be a triangle inscribed (O). AA, BB , CC are altitudes. AA intersect (O)at second point D. E is a point on AB such that BE ⊥ OA. DB intersect (O) at second point F .BF intersects AE at K . Prove that K is midpoint of BC .

Problem 23. Let P be a point inside triangle ABC such that P B + AC = P C + AB. P B, P C intersect AC , AB at Y , Z . M is midpoint of Y Z .

a) Prove that radical axis of incircles of triangles P ZB and P Y C pass through M .b) Prove that radical axis of incircles of triangles Y BC and ZBC pass through M .

Problem 24. Let ABC be a triangle inscribed circle (O). K a is reflection of O through BC . (K a)is circle center K a and passes through B, C . (Oa) touchs AB, AC and touchs (K a) externally at A

( Oa is inside triangle ABC ). Similarly we have B, C . Prove that AA, BB , CC are concurrent.

Problem 25. Let ABC be a triangle. A circle (ω) pass through B, C intersect AC , AB at E , F .BE intersects C F at I . AI intersect circle (ω) at Z (the point Z is inwardly to triangle AB C ). BE ,CF intersect circumcircle (ACF ), (ABE ) at M , N , resp ( M , N are outwardly to triangle ABC ).Let the points M ≡ AB ∩ ZM , N ≡ AC ∩ ZN and K ≡ M N ∩M N . Prove that KM = K N .

Problem 26. Let D is a point on BC of triangle ABC . P is a point on AD. BP , CP intesects circumcircles (ACP ), (ABP ) at K , L, resp. BP , CP intesects circumcircles (P BD), (P CD) at M , N , resp. Prove that midpoints of the segments KL, M N , BC are collinear.

Problem 27. Let ABC be a triangle. P , Q are the points on BC such that circumcircle (ABQ),(ACP ) touchs AC , AB, resp. AP , AQ intersect circumcircle (ABC ) at M , N , reps. Circumcircle (AQM ), (AP N ) intersect BC at D, E , resp. Prove that BD = C E .

Problem 28. Let AB be a segment and P , P are two arbitrary points. I is a point on AB. M ,N lie on P A, P B, resp, such that IM P B, M N AB. M , N lie on P B, P A, resp, such that IM P A, M N AB. K lies on circumcircle (IM N ) such that M K IN . L lies on circumcircle ( IM N ) such that M L I N . Prove that I , K , L are collinear.


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Problem 29. Let ABC be a triangle. The circle (ω) which passes though A, C and tangent to AB,meets BC at D. d passes through A is isogonal line of AD. X is a point on d. Y is a point on AD.BX , CY intersects AC , AB at E , F , resp. Z lies on EF such that Y Z AC . AZ intersects XC at T . Prove that T lies on (ω).

Problem 30. Let ABC be a triangle. The circle (ω) which passes though A, C and tangent to AB,

meets BC at D. d is a line which passes through A. X is a point on d. Y is a point on AD. BX ,CY intersects AC , AB at E , F , resp. XC intersects (ω) at T ( T ≡ C ). AT intersects EF at Z .Prove that Z lies on a fix line which passes though Y when X moves on d.

Problem 31. Let ABC be a triangle incircle (I ) touchs BC , CA, AB at A1, B1, C 1. A2, B2, C 2lies on IA1, IB1, IC 1 such that IA2. IA1 = IB2. IB1 = IC 2. IC 1. O is circumcenter of triangle ABC . OA2, OB2, OC 2 cuts circumcircle (AIA2), (BI B2), (CIC 2) again at A3, B3, C 3. Prove that AA3, BB3, CC 3 are concurrent on point P line OI .

Problem 32. Let ABC be a triangle with altitude AD and circumcircle (O). P is a point on AD.Circle with diameter AP intersects AB, AC , (O) again at C , B, Q, resp. Prove that P Q, BB ,CC are concurrent.

Problem 33. Let ABCD be quadrilateral. AC cuts BD at O. Let K , L be circumcenter of triangle OAD, OBC , resp. Circumcircle (OAB), (OC D) intersect again at I . Prove that IK ⊥ BC ⇐⇒IL ⊥ AD.

Problem 34. Let ABC be a triangle and P , Q be two isogonal conjugate points. A, B, C are reflections of P through BC , CA, AB, resp. QA, QB, QC cuts BC , CA, AB at D, E , F , resp.Prove that AD, BE , CF are concurrent.

Problem 35. Let ABC be a triangle and ABC is cevian triangle of a point P . Intersections of BC , B C ; C A, C A; AB, AB are collinear on d. DEF is cevian triangle of a point Q with respect to ABC . BE , C F cut d at M , N , resp. M B cuts N C at R. AR cuts BC at T . T M , T N cut

A

C

, A

B

at K , L, resp. Prove that B

K , C

L and RT are concurrent.

Problem 36. Let ABC be a triangle and a point P . A line pass through P intersect circumcircle (P BC ), (P CA), (P AB) again at P a, P b, P c, resp. Let a, b, c, be tangets of circumcircle (P BC ),(P CA), (P AB) at P a, P b, P c, resp. Prove that the circumcircle of the triangle determined by the lines a, b, c is tangent to the circumcircle (ABC ).

Problem 37. Let p, q, r be three lines concur at O, a 1, a 2, a 3 be three three lines concur at A. pintersect a 1, a 2, a 3 at P 1, P 2, P 3, q intersect a 1, a 2, a 3 at Q1, Q2, Q3, r intersect a 1, a 2, a 3 at R1,R2, R3, resp. P 3Q2 cuts r at K . L is a point such that (P 2Q2AL) = −1. Assume that Q1R2, P 2R3and OA are concurrent. Prove that R1P 2, Q2R3 and KL are concurrent.

Problem 38. Let ABCD be cyclic quadrilateral with circumcircle (O). l is a tangent of (O). l 1, l 2,l 3, l 4 are reflections of l through AB, BC , CD, DA, resp. Prove that there are four cyclic points in six intersections of l 1, l 2, l 3, l 4 and its circumcenter lies on a fixed circle when l move.

Problem 39. Let ABCD be cyclic quadrilateral. P is a point on plane. l is a line pass through P .P 1, P 2, P 3, P 4 are intersections of l with circumcircle (P AB), (P BC ), (P CD), (P DA), resp. l 1,l 2, l 3, l 4 is tangent of circumcircles (P AB), (P BC ), (P CD), (P DA) at P 1, P 2, P 3, P 4, resp. Prove that there are four cyclic points in six intersection of l 1, l 2, l 3, l 4.


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Problem 40. Let ABCD be cyclic quadrilateral inscribed (O). AC cuts BD at P . M , L lie on AD,N , K lie on BC such that M NK L is cyclic and M K , NL pass through P . Prove that circumcenter O∗ of (MNKL) lies on OP .

Problem 41. Let ABC be a triangle and a circle pass thourgh B, C cuts AB, AC at F , E , resp.M is midpoint of BC . M T 1, M T 2 are tangent of (AEF ) at T 1, T 2, resp. I , K are midpoint of M T 1,

M T 2, resp. IK cuts BC at T . EF cuts a line pass through A parallel to BC at S . Prove that ST is tangent to circumcircle (AEF ).

Problem 42. Let AB, CD, EF be chords of circle (O) such that segment EF cuts segments AB,CD at M , N and A, C are in the same side with EF . (O1) touches M E , M B and (O) internally,(O2) touches N F , N D and (O) internally. P , Q are contact points of EF with (O1), (O2), R, S are contact points of (O) with (O1), (O2). Prove that angle bisector of ∠P O1R, ∠QO2S , ∠O1OO2 are concurrent.

Problem 43. Let ABC be a triangle inscribed (O). Bisector of ∠BAC intersects (O) again at D.P is a point on AD. Q lies on AD such that AP . AQ = AB. AC . E is isogonal conjugate of D

with respect to triangle P BC . M is midpoint of AQ. Prove that M E always passes throuh centroid G of triangle ABC .

Problem 44. Let AB C be a triangle inscribed (O). Bisector of ∠BAC intersects (O) again at D. P is a point on AD. E is isogonal conjugate of D with respect to triangle P BC . Q, R are projections of P on AC , AB, resp. BQ cuts CR at F . Prove that EF passed through midpoint of QR.

Problem 45. Let ABC be triangle inscribed circle (O). XY Z be pedal triangle of a point P with respect to triangle ABC . P A cuts (O) again at D. DE is a chord of (O) and perpendicular to BC .I is a midpoint of DE . P I cuts BC at F . F A cuts parallel to P A through the point X at T . Prove midpoint of XT lies on Y Z .

Problem 46. Let ABC be triangle and XY Z is pedal triangle of a point P with respect to ABC .X is reflection of X through Y Z , Y Z cuts BC at T . P X cuts Y Z at S . Circumcircle (AST ) cuts T X again at M . O is circumcenter of triangle ABC prove that M , A, O are collinear.

Problem 47. Let ABC be a triangle inscribed circle (O). Let P is a point on perpendicular bisector of BC . D is middle of the arc BC , not containing A. M , L are projections of P on AC , AB, resp.DM intersects BC , (O) at X , Y . DL intersects BC , (O) at Z , T . Prove that X , Y , Z , T are concyclic.

Problem 48. Let ABC be a triangle. P , P are two isogonal conjugate point with respsect to ABC .K , K are projection of P and P on line BC , resp. AH is altitude of ABC . A1, A2 ∈ AH such

that AA1 = P K , AA2 = P K . P A1, P A2 intersect line BC at P , Q. Prove that BP = C Q.

Problem 49. Let ABC be a triangle and XY Z is pedal triangle of a point P with respect to ABC .P is isogonal conjugate of P . ( O, R) is circumcircle of triangle XY Z . C is circle ( P , 2R). Ray Y P , ZP intersects C at M , N , resp. M N cuts Y Z at R. T is projection of R on P A. Prove that T is inversion of A with respect to C .


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Problem 50. T is the midpoint of side AB of the convex quadrilateral ABCD. The circle k through C , D intersect AB at X , Y such that and T is midpoint of X Y . K and L are the intersection points of AD and BC respectively with k. M and N are the intersection points of AC and BD respectively with KL. P and Q are the intersection points of DM and CN respectively with the segment AB.Prove that AP = B Q.

Problem 51. Let ABC be a triangle inscribed (O) and point P . P A, P B, P C intersect (O) again at X , Y , Z . X , Y , Z are reflections of X , Y , Z through OP . Prove that AX , BY , CZ and OP are concurrent.

Problem 52. Let ABC be a triangle inscribed (O). P aP bP c is pedal triangle of point P such that circumcenter of (P aP bP c) lies on OP . P A, P B, P C intersect (O) again at X

, Y , Z . X , Y , Z are reflections of P through BC , CA, AB. XX , Y Y , ZZ intersect (O) again at A, B, C . Let P aP

bP

c be pedal triangle of P with respect to triangle ABC . Prove that the triangles P aP bP c and

P aP

bP

c have the same circumcircle.

Problem 53. Let ABC be a triangle and the points M , N , P . BC , CA, AB intersect M N at A,B, C . P A intersect M N at A1. Let A2 be a point such that cross ratio ( M N C

A1) =( N M BA2).

Defined similarly B2, C 2. Prove that AA2, BB2, CC 2 are concurrent. Let ABC be a triangle and a point P . A1B1C 1 is pedal triangle of P . P

∗ is isogonal conjugate of P . A2B2C 2 is pedal triangle of P ∗. Q is a point on line P P ∗. A2Q, B2Q, C 2Q cut cirumcircle (A1B1C 1) again at A3, B3, C 3,respectively.

a) Prove that A1A3, B1B3, C 1C 3 are concurrent on line P P ∗.

b) Prove that AA3, BB3, CC 3 are concurrent.

Problem 54. Let ABC be triangle. (O) is a circle which passes through B, C . AB, AC cut (O)again at F , E . BE cuts CF at D.

a) Prove that tangent at E , F of (O) and AD are concurrent at T b) DA cuts EF at G, BG cuts T C at M , CG cuts T B at N . Prove that M , N lie on (O).

Problem 55. Let ABC be a triangle. Incircle (I ) touches BC , CA, AB at D, E , F . M is a point on circle center A which passes though E , F .

a) Prove that pedal triangle XY Z of M with respect to triangle DEF is right triangle.b) DM cuts IA at K . M I cuts EF at T . Prove that K lies on circumcircle (DEF ) if only if T

lies on circumcircle (XY Z ).c) M ∗ is isogonal conjugate of M with respect to triangle DEF . Prove that M ∗ always lies on

fixed circle.

Problem 56. Let ABC be a triangle and point P . ABC is pedal triangle of P with respect totriangle ABC . O is circumcenter of triangles ABC , ( O) is circumcircle of triangle ABC . P A,

P B

, P C

intersects ( O

) again at A1, B1, C 1, respectively. Assume that P , O, O

are collinear.Prove that circumcirles (P AA1), (P BB1), (P CC 1) have a common point other than P .

Problem 57. Let AB C be a triangle with circumcircle (O). A circle (K ) pass though B , C intersects AB, AC at F , E , respectively. O1, O2 are circumcenter of triangles ABE , ACF , respectively. (L)is circumcircle of triangle KO1O2. P is point on (L). The line passes though P and perpendiculer toOP intersects (O) at B, C . Prove that nine −point center of triangle ABC always lies on a fixed circle (J ) and LJ ⊥ E F .


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Problem 58. Let ABC be a triangle. D is a point on BC such that ∠BAD = ∠ACB. E is a point on circumcircle (ACD) such that DE AC . P is a point on AE . P B cuts DE at F . Prove that AF and CP intersect on circumcircle (ACD).

Problem 59. Let ABCDEF be a hexagon with circumcircle (O) and incircle (I ). P 1, P 2 are twopoints on OI . AP 1 cuts (O) again at A1. A1P 2 cuts (O) again at A2. Similarly we have B1, B2, C 1,

C 2, D1, D2, E 1, E 2, F 1, F 2. Prove that A2D2, B2E 2, C 2F 2, OI are concurrent.

Problem 60. Let ABC be a triangle and (K ) is a circle pass through B, C . AC , AB cuts (K )again at E , F , resp. I is circumcenter of triangle AE F . Ray I A cuts the circle ( I , IK ) at T . Prove that KT is parallel to angle bisector of ∠BAC .

Problem 61. Let ABC be a triangle and (K ) is a circle pass through B, C . AC , AB cuts (K )again at E , F . BE cuts CF at H . (O) is circumcircle of triangle ABC .

a) Prove that HK , AO intersects at point A on (O).b) Circumcircles (BF H ), (CEH ) cut BA, CA again at M , N , resp. (BF H ) cuts (CEH ) again

at D. Prove that A, H , D are collinear and M , N , K , D are concyclic.

c) Let I be circumcenter of triangle KM N . Prove that IH and tangent at B, C of (O) are concurrent.

Let ABC be a triangle and ABC is pedal triangle of a point P . AA cuts P B, P C at M , N .The lines pass through M , N and parallel to BC cut AB, AC at K , L, respectively. Prove that K ,P , L are collinear.

Problem 62. Let ABC be a triangle. (K ) is a circle passing through B, C . (K ) cut AC , AB again at E , F . BE cuts CF at G. H is projection of K on AG. L is circumcenter of triangle HEF . N ,P lie on AC , AB, resp such that LN AB , LP AC . The line passing through N parallel to BE cuts the line passing through P parallel to CF at T . S is midpoint of AG. Prove that ST passes thought fixed point when (K ) varies.

Problem 63. Let ABC be a triangle with circumcircle (O). (K ) is a circle passing through B, C .(K ) cuts CA, AB again at E , F . BE cuts CF at H K .

a) Prove that H K K and AO intersect on (O).b) OK is isogonal conjugate of H K with respect to triangle ABC . Prove that OK lies on OK .c) Let L, N be the points on CA, AB, resp such that OK L BE , OK N CF . Prove that

LN B C .d) The line passing through N parallel to BE cuts the line passing through L parallel to CF at

P . Prove that P lies on AH K .e) Q, R lie on BE , CF , resp such that P Q AB , P R AC . Prove that QR B C .

f) Prove that NQ, LR and AH K are concurrent.

g) D is projection of K on AH K . Prove that DK , EF , BC are concurrent.h) Prove that KN ⊥ B E , KL ⊥ C F .i) Prove that nine points D, E , F ; P , Q, R; K , L, N lie on a circle (N K ).

j) Prove that N K is midpoint of P K and KN K is parallel to AO.k) Prove that H K , N K , O are collinear.


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Problem 64. Let ABC be a triangle with circumcenter O and a point P . P A, P B, P C cut BC ,CA, AB at A1, B1, C 1. A2B2C 2 is pedal triangle of P , assume that AA2, BB2, CC 2 are concurrent.Let A3 ∈ B1C 1, B3 ∈ C 1A1, C 3 ∈ A1B1 such that P A3 ⊥ BC , P B3 ⊥ CA, P C 3 ⊥ AB. Prove that AA3, BB3, CC 3 and P O are concurrent.

Problem 65. Let ABC be triangle. A circle (K ) passing through B, C cuts CA, AB at E , F . BE

cuts CF at G. AG cuts BC at H . L is projection of H on EF . M is midpoint of BC . M K cuts circumcircle (KEF ) again at N . Prove that ∠LAB = ∠N AC .

Problem 66. Let ABC be a triangle and A1B1C 1 is pedal triangle of a point P with respect totriangle ABC . T is a point on circumcircle (A1B1C 1). is line passing though T and perpendicular to P T . A2, B2, C 2 lie on such that P A2 ⊥ P A, P B2 ⊥ P B, P C 2 ⊥ P C . Prove that AA2, BB2,CC 2 are concurrent.

Problem 67. Prove that in a triangle, orthocenter, symmedian point of anticomplementary triangle,third Brocard point are collinear. (Third brocard point is isotomic conjugate of symmedian point).

Problem 68.

Let ABC

be triangle with ∠A

= 60

◦

. O

, K

are circumcenter and symmedian point of triangle ABC , resp. OK cuts AB, AC at F , E . O, K are circumcenter and symmedian point of triangle AEF , resp. Prove that OK , OK and BC are concurrent.

Problem 69. Let ABC be a triangle with circumcircle (O). D, E lie on (O). Circle (C 1) pass through A, D and tangent to AC . Circle (C 2) pass through A, E and tangent to AB. (C 1) cuts (C 2)again at P . Prove that AP , BD, CE are concurrent.

Problem 70. Let ABC be triangle. P is a point and D, E , F are projections of P on lines BC ,CA, AB. P B cuts DE at M . P C cuts DF at N . M N cuts P A at Q. P B, P C cut the line passing through Q and perpendicular to P A at K , L, resp.

a) Prove that the line passing through K perpendicular to P C , the line passing through L perpen-dicular to P B and P A are concurrent at T .

b) U , V are reflections of T through P C , P B, resp. Prove that U , V , K , L, P lie on circle (O1).c) U V cuts K L at S . (O2) is circumcircle of triangle S KU , (O3) is circumcircle of triangle S LV .

Circle (O2) and (O3) intersects again at W . Prove that O1, O2, O3, W , P lie on a circle center J .d) Prove that K , L, J are collinear.

Problem 71. Let ABC be triangle and P is a point. E , F are projections of P on lines CA, AB,resp. Q is isogonal conjugate of P with respect to triangle ABC . F P cuts EQ at K , EP cuts F Qat L. Prove that circumcircle (P EK ) and (P F L) intersects again on line P A.

Problem 72. Let ABC be triangle. A circle passing through B, C cuts CA, AB at E , F . Tangent

at E , F of circumcircle (AEF ) intersects at K . M , N is midpoints of KE , KF . M N cuts CA,AB at P , Q, resp. Prove that A, P , Q, K are concyclic.

Problem 73. Let ABC be a triangle. A circle passing through B, C cuts CA, AB at E , F , resp.BE cuts CF at H . M , N are on AB, AC such that M N passing through H . K is in BH such that M K is tangent of circumcircle (F M H ). L is in CH such that NL is tangent of circumcircle (EN H ). Prove that KL B C .


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Problem 74. Let ABC be triangle with circumcircle (O). P is a point and P 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 N have the same A−symmedian.

Problem 75. Let ABC be a triangle with point P . Circumcircle (P AB) cuts AC again at E .Circumcircle (P CA) cuts AB again at F . M , N are midpoints of BC , EF , resp. Q is isogonal

conjugate of P with respect to triangle ABC . Prove that M N AQ.

Problem 76. Let AB C be triangle with circumcircle (O) and a point D. (O1), (O2) are circumcircles 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) D B 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 77. Let ABC be triangle and a point P . A1B1C 1 is pedal triangle of P with respect totriangle ABC . A2B2C 2 is circumcevian triangle of P . A3B3C 3 is pedal triangle of P with respective to triangle A2B2C 2. Prove that A1B1C 1 and A3B3C 3 are persective if only if A1B1C 1 and A2B2C 2

are perspective.

Problem 78. Let ABC be triangle with circumcenter O. Tangent at A of circumcircle (ABC ) cuts BC at T .

a) Circle ω ( T , T A) cuts (ABC ) again at D. Prove that AD is symmedian of triangle ABC .(Actually, ω is A− apollonius circle of triangle ABC )

b) A line d passes though O cuts (ω) at M , M . Prove that intersecton of tangents at M , M of (ω) lies on symmedian of triangle ABC .

c) Let N , N is isogonal conjugate of M , M with respect to triangle ABC . Prove that tangents at N , N of circumcircle ( AN N ) intersects on line BC .

d) Prove that midpoint of N N lies on nine −points circle of triangle ABC .

Problem 79. Let ABCD be circumscribed quadrilateral with incenter I . O1, H 1 is circumcenter

and orthocenter of triangle IAB. K 1 is a point such that −−−→O1K 1 = k

−−−→O1H 1, k is a fixed constant, d 1 is

the line passing through K 1 and perpendicular to AB. Similarly we get d 2, d 3, d 4. Prove that d 1, d 2,d 3, d 4 form circumscribed quadrilateral.

Problem 80. Let ABC be a triangle with circumcircle (O). P is a point on line BC outside (O). T is a point on AP such that BT , CT cuts (O) again at M , N , resp, then M N P A. Q is reflection of P through M B, R is reflection of P through N C . Prove that QR ⊥ B C .

Problem 81. Let ABCD be cyclic quadrilateral. d is perpendicular bisector of BD. P is a point

on d. Q

is reflection of P

through bisector of angle ∠BAD

. R

is reflection of P

through bisector of angle ∠BC D. Prove that AQ, CR and d are concurrent.

Problem 82. Let ABC be triangle with circumcircle ( O, R). P , P ∗ are two isogonal conjugate points with respect to triangle ABC . Q is reflection of P through BC . AP , AP ∗ cut (O) again at D, D. DQ cuts (O) again at E . EP ∗ cuts (O) again at E . Prove that AE D E .


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Problem 83. Let ABC be a triangle with circumcircle (O) and a point P . AP cuts (O) again at D. E is on (O) such that DE ⊥ B C . EP cuts BC at F and (O) second time at G. Q is a point on

AG such that F Q AP . Prove that tan P AB

tan P AC =

tan F QB

tan F QC .

Problem 84. Let ABC be a triangle with circumcircle (O). D, E are on (O). DE cuts BC at T .Line passes though T and parallel to AD cuts AB, AC at M , N . Line passes though T and parallel to AE cuts AB, AC at P , Q. Perpendicular bisector of M N , P Q cut perpendicular bisector of BC at X , Y , resp. Prove that O is midpoint of XY .

Problem 85. Let ABCD be a cyclic quadrilateral. Circle pass through A, D cuts AC , DB at E ,F . G lies on AC such that BG DE , H lies on BD such that CH AF . AF ∩ DE ≡ X ,DE ∩CH ≡ Y , CH ∩GB ≡ Z , GB∩AF ≡ T . M lies on AC . N lies on BD such that MN AB,P lies on AC such that N P BC , Q lies on BD such that P Q CD. d M passes though M , d P passes though P such that d M d P DE . d Q passes though Q, d N passes though N such that d Q d N AF . d Q∩ d M ≡ U , d M ∩ d N ≡ V , d N ∩ d P ≡ W , d P ∩ d Q ≡ S . Prove that XU , ZW , SY , T V are concurrent.

Problem 86. Let O be circumcenter of triangle ABC . D is a point on BC . (K ) is circumcircle of triangle ABD. (K ) cuts OA again at E .

a) Prove that B, K , O, E are concyclic.b) (K ) cuts AB again at F . G is on (K ) such that OG E F . GK cuts AD at S . SE cuts BC

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

Problem 87. Let ABC be triangle and P , Q are two isogonal conjugate points with respect totriangle ABC . Prove that circumcenter of the triangles P AB, P AC , QAB, QAC are concyclic.

Problem 88. Let ABCD be a parallelogram. (O) is circumcircle of triangle ABC . P is a point on

BC . K is circumcenter of triangle P AB. L is in AB such that KL ⊥ BC . CL cuts (O) again at M . Prove that M , P , C , D are concyclic.

Problem 89. Let ABC be a triangle and P , Q are two arbitrary point. A1B1C 1 is pedal triangle of P with respect to triangle ABC . A2, B2, C 2 are symmetric of Q through A1, B1, C 1, respectively.A3, B3, C 3 are reflection of A2, B2, C 2 through BC , CA, AB , respectively. Prove that Q, A3, B3,C 3 are concyclic.

Problem 90. Let ABC be a triangle with circumcircle (O) and a point P . P A, P B, P C cuts (O)again at A1, B1, C 1. A2, B2, C 2 is pedal triangle of P with respect to triangle AB C . H is orthocenter of triangle ABC . A3, B3, C 3 are symmetric of H thourgh A2, B2, C 2 respectively. Prove that A1A3,

B1B3, C 1C 3 are concurrent at point T lies on (O).

Problem 91. Let ABCD be cyclic quadrilateral. The lines passing through midpoint of a side and perpendicular to opposite side are concurrent at point M . m, n are two perpendicular lines passing through M . m cuts AB, BC , CD, DA, CA, BD at E , F , G, H , I , J , respectively. n cuts AB,BC , CD, DA, CA, BD at X , Y , Z , T , U , V . P , Q are midpoints of XE , GZ , K , L are midpoints of F Y , HT , N , P are midpoints of U I , J V , respectively. Prove that P Q, KL, N P are concurrent.


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Problem 92. Let ABCDE be bicentric pentagon with incircle (I ) and circumcircle (O). M , N , P ,Q, R are intersections of diagonals a figure. Constructed circles (K 1), (K 2), (K 3), (K 4), (K 5) as in

figure. Prove that XM , Y N , ZP , T Q, U R are concurrent at a point S on OI .

Problem 93. Let ABC be a triangle and D is a point on BC . M is a point on AD. The Line passing through M and parallel to BC cuts CA, AB at E , F , respectively. The line passing through

E and parallel to AB cuts the line passing through F and parallel to CA at P . N is a point on line P M . N B cuts P F at K , N C cuts P E at L, CK cuts BL at Q. Prove that P Q AD.

Problem 94. Let ABC be a triangle and A−excircle touches BC at D. d is a line passing through D. d cut CA, AB at E , F , respectively. M is E −excenter of triangle DC E , N is F −excenter of triangle DBF , P is incenter of triangle AEF . Prove that A, M , N , P are concyclic.

Problem 95. Let ABCD be cyclic quadrilateral with circumcircle (O). AC cuts BD at I . E , F ,G, H are incenters of triangles I AB, IBC , ICD, IDA, respectively. (K ), (L), (M ), (N ) are circles tangent to lines IA, IB; IB, IC ; IC , ID; ID, IA and tangent to (O) internally at X , Y , Z , T ,respectively.

a) Prove that X , E , G, Z and Y , F , H , T are concyclic on (O1) and (O2), respectively.

b) Prove that O lies on radical axis of (O1) and (O2).

Problem 96. Let ABCDEF be bicentric hexagon with incircle (I ) and circumcircle (O).a) Prove that AD, BE , CF are concurrent at point K on OI .b) Constructed circles (K 1), (K 2), (K 3), (K 4), (K 5), (K 6) as in figure. Let G, H , J , L, M , N

be incenters of triangles KAB, KBC , KC D, KD E , KE F , KF A, respectively. Prove that X , G,L, T ; Y , H , M , U ; Z , J , N , V are concyclic on circles (O1), (O2), (O3), respectively.

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

Problem 97. Let ABC be a triangle with circumcenter O. P , Q are two isogonal conjugate with respect to triangle such that P , Q, O are collinear. Prove that four nine −point circles of triangles ABC , AP Q, BP Q, CP Q have a same point.

Problem 98. 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 Fe, G be Feuerbach points of triangle ABC and M N P . Prove that OI passes through midpoint of Fe G.

Problem 99. Let P be a point on A angle bisector of triangle ABC . D, E , F are projections of P on BC , CA, AB. Circumcircle of triangle AEF intersects DE , DF again at M , N , resp. AM ,AN cut BC at P , Q. Prove that D is midpoint of P Q.

Problem 100. Let ABC be a triangle and a point P . Let D, E , F be projection of P on BC , CA,AB. Circumcircle of triangle AEF intersect DE , DF again at M , N , resp. AM , AN intersect BC

at Q, R. Prove that P RP Q

= P E P F

.

Problem 101. Let (O1) and (O2) be two circles and d is their radical axis. I is a point on d. IA,IB tangent to (O1), (O2) ( A ∈ (O1), B ∈ (O2)) and A, B have same side with O1O2, 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 .


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Problem 102. Let ABC be a triangle with circumcircle (O). P is a point. P T is diameter of circumcircle triangle P BC . P T , P C , P B cut (O) again at D, E , F , respectively. DQ is diameter of (O). AQ cuts EF at S . Prove that P D ⊥ S T .

Problem 103. Let ABC be a triangle P is a point such that pedal triangle DEF of P is also cevian triangle. P B, P C cut EF at X , Y , repectively. Prove that DP is angle bisector of ∠XDY .

Problem 104. Let ABC be a triangle with circumcircle (O) and P , Q are two isogonal conjugate points. A1, B1, C 1 are midponts of BC , CA, AB, resp. P A, P B, P C cut (O) again at A2, B2, C 2,resp. A2A1, B2B1, C 2C 2 cut (O) again at A3, B3, C 3. A3Q, B3Q, C 3Q cut BC , CA, AB at A4,B4, C 4. Prove that AA4, BB4, CC 4 are concurrent.

Problem 105. Let ABCD be circumscribed quadrilateral and M is its Miquel point. (M ) is a circle center M . Let A, B, C , D invert A, B, C , D through (M ), respectively. Prove that ABC D is circumscribed quadrilateral.

Problem 106. Let ABC and ABC be two triangles inscribed circle (O). Prove that orthopoles of BC , C A, AB with respect to triangle ABC and orthopoles of BC , CA, AB with respect totriangle ABC lie on a circle.

Problem 107. Let ABC be a triangle inscribed circle (O). P is a point on (O). Prove that Steiner line of P with respect to triangle ABC and orthotransversals of P with respect to triangle ABC intersect on Jerabek hyperbola of triangle ABC .

Problem 108. Let ABC be a triangle and a point P . DEF is cevian triangle of P with respect totriangle ABC . M , N are on line EF such that BM C N . Let Q be a point such that PBQC is a prallelogram. BQ cuts AC at K , CQ cuts AB at L.

a) Prove that M K , N L and BC are concurrent at a point T .b) Prove that if BM C N P A then T is midpoint of BC .

Problem 109. Prove that orthopole of orthotransversal of a point that lies on nine points circle of

a triangle, also is that point.

Problem 110. Let ABC be a triangle inscribed an Ellipse (E ) with center O is midpoint of BC .M , N lie on (E ) such that OM AB, ON AC and M , N are different side of A with BC . Let AM cuts ON at P , AN cuts OM at Q. Prove that line P Q bisects the area of triangle ABC .

Problem 111. Let ABC be a triangle inscribed circle (O) and orthocenter H . P is a point on (O).d is Steiner line of P . M is Miquel point of d with respect to triangle ABC . Prove that M , P , H are collinear iff OP ⊥ d.

Problem 112. Let ABC be triangle inscribed (O). (Oa) is A−mixtilinear incircle of ABC . Circle (ωa) other than (O) passing through B, C and touches (Oa) at A

. Similarly we have B, C . Prove

that AA

, BB

, CC

are concurrent.Problem 113. Let AB C be a triangle with orthocenter H and circumcenter O. (ω) is a circle center O and radius k. (K ), (L) are two circle passing through O, H and touch (ω) at M , N .

a) Prove that (K ), (L) have the same radius.b) Prove that M N passes through H .c) Prove that circle (Ω) center H radius k also touches (K ) and (L) at P , Q and P Q passes

through O.


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Problem 114. P , Q are antigonal conjugates with respect to ABC . Then 9−point circles of AP Q, BP Q, CP Q are tangent.

Problem 115. Let P , Q be two antigonal conjugates with respect to triangle ABC . Circumcircle of triangle AP Q, BP Q, CP Q cut circumcircle of triangle ABC again at A, B, C , respectively.Prove that AA, BB , CC are concurrent.

Problem 116. Let ABC be a triangle, orthocenter H and a point P . Let ABC be pedal triangle of P . (E ) is circumellipse of triangle ABC with center is midpoint of P H . Prove that orthopole of any line passing though P lies on (E ).

Problem 117. Prove that two orthotransversals of two antigonal conjugate points with respect to a triangle are parallel.

Problem 118. Let AB C be a triangle with circumcenter O and a point P . d is a line passing though P and perpendicular to OP . d cuts circumcircle of triangle P BC , P CA, P AB again at X , Y , Z ,respectively. A circle center O cuts OA, OB, OC at A, B, C , respectively. Prove that AX , BY ,C Z

are concurrent.

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Tran Quang Hung - Red Geometry