Tran Quang Hung - Red geometry

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Red geometry by Tran Quang Hung Problem 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 MN with P A. K is point on BC such that ∠MKN = 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, A0 B 0 C 0 is pedal triangle of P , through A0 , B 0 , C 0 draw da k P A, db k P B, dc k P C, H is orthocenter of triangle ABC. da ∩ HA = A00 , db ∩ HB = B 00 , dc ∩ HC = C 00 , circumcircle of A0 B 0 C 0 intersects nine−point circle of triangle ABC at F , F 0 prove that the circles with diameters (A0 A00 ), (B 0 B 00 ), (C 0 C 00 ) are through F or F 0 . Problem 3. Let ABC be a triangle and A0 B 0 C 0 is pedal triangle of an arbitrary point P , let P B 0 , P C 0 intersect C 0 A0 , A0 B 0 ay Y , Z, resp. BY , CZ intersect CA, AB at E, F , resp. P B intersects EF at D. Prove that A0 , B 0 , 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 ABC, DP ∩ AO = O 0, Y , Z are projections of O 0 on AB, AC. Prove that ZY HP is isoceles trapezoid. Problem 5. Let P is a point on altitude of triangle ABC and B 0 , C 0 are projections of P on AC, AB. a) Prove that B 0 , C, B, C 0 are concyclic with center I. b) Let AI ∩ BC = J and K is projection of J on line B 0 C 0 , Q is a point on line AJ. M, N are midpoint of BC 0 and CB 0 . The perpendicular to line QB, QC through M, N, resp intersect at point H. Prove that HB = HC. 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 0 , Y 0 , Z 0 , T 0 , U 0 , V 0 . Let E, F , K, L, I, J be midpoints of XX 0 , Y Y 0 , ZZ 0 , T T 0 , UU 0 , V V 0 . 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 A0 , B 0 , C 0 be center of these rectangles (cyclic with A, B, C), respectively. a) Prove that AA0 , BB 0 , CC 0 are concurrent. b) Let A00 , B 00 , C 00 are reflections of A0 , B 0 , C 0 through BC, CA, AB, resp. Prove that AA00 , BB 00 , CC 00 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 , `d parallel 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.

Tran Quang Hung - Red geometry

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Problem 9. Let (O1 ), (O2 ) be two orthogonal circles, and P is a point on (O2), draw two tranversal P MN and P EF (M, N, E, F ∈ (O1 ), this means P , M, N and P , E, F are collinear). Let ME ∩ NF = K, P O1 ∩ (O2 ) = L then P L ⊥ LK. Problem 10. Three medians AA0 , BB 0 , CC 0 of triangle ABC are concurrent at G. Take A1 , −−→ −−−→ A2 ∈ AA0 such that A1 A2 = 21 AA0 similarly we have B1 , B2 , C1 , C2 . Draw two lines through A1 , A2 and perpendicular to AA0 , 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 A0 B 0 C 0 be a Cevian triangle of point P . P A0 , BC intersect circumcircle (A0 B 0 C 0 ) at {A0 , A1 } and {A0 , A2 }, resp. A1 A2 intersect B 0 C 0 at H. Tangent of (A0 B 0 C 0 ) at A2 intersects B 0 C 0 , AB, AC, AH at I, J, K, L, resp. IA intersect A1 A2 , A2 B 0 , A2 C 0 at I, M, N, resp. Prove that NJ, KL, MI are concurrent. Problem 12. Let H be orthocenter of triangle ABC and A1 B1 C1 is pedal triangle of H wrt ABC, A2 B2 C2 is pedal triangle of H wrt A1 B1 C1 . Take A3 , B3 , C3 on ray HA2 , HB2 , HC2 such that HA3 = HB3 = HC3 . a) Prove that radical center of circles (A1 , A1 A3 ), (B1 , B1 B3 ), (C1 , C1 C3 ) lie on Euler line of triangle ABC. b) Prove that radical center of circles (A, AA3 ), (B, BB3 ), (C, CC3 ) lie on Euler line of triangle ABC. Problem 13. Let ABCDEF be a hexagon inscribed circle (O) such that AD, BF , CE are concurrent. 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 A0 , B 0 , C 0 , D 0 , E 0 Prove that A0 D 0 , B 0 F 0 , C 0 E 0 are concurrent. Problem 14. Let ABCDEF be a cyclic hexagon and point P . Let A0 , B 0 , C 0 , D 0 , E 0 , F 0 be circumcenter of triangles P AB, P BC, P CD, P DE, P EF , P F A. Prove that A0 D 0 , B 0 E 0 , C 0 F 0 are concurrent. Problem 15. Let ABC be a triangle and A0 B 0 C 0 be Cevian triangle of point O. M, N, P are midpoint of AB, BC, OA. d is a line through B. MB 0 , MP , MN intersect d at E, F , G, resp. B 0 C 0 ∩ BC = {I}. IF ∩ GA0 = {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 A0 B 0 C 0 be cevian triangle of P . Through midpoint of P A draw line parallell to BC, it cuts 0 0 B C at Q. OB ∩ A0 C 0 = {B1 }, OC ∩ A0 B 0 = {C1 } . Prove that P Q k B1 C1 Problem 17. Let ABC be a triangle and point P , A1 B1 C1 is cevian triangle of P . A2 , B2 , C2 are midpoints of BC, CA, AB. Q is an arbitrary point. QA2 , QB2 , QC2 intersect P A, P B, P C at A3 , B3 , C3 , resp. Let da , db , dc be the lines through A, B, C and parallel to QA2 , QB2 , QC2 , resp. da , db , dc intersect BC, CA, AB at A4 , B4 , C4 . Let A5 , B5 , C5 be the point such that A4 , B4 , C4 be midpoints of AA5 , BB5 , CC5 , reps. Let Ga , Gb , Gc be centroid of triangle A3 BC, B3 CA, C3 AB, assume that AGa , BGb , CGc are concurrent, prove that A5 A1 , B5 B1 , C5 C1 are concurrent.

Tran Quang Hung - Red geometry

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Problem 18. Let A0 B 0 C 0 be a pedal triangle of arbitrary point P with respect to triangle ABC. P ∗ is isogonal conjugate of P with respect to triangle ABC. Rp is circumradii of triangle A0 B 0 C 0 . The rays [B 0 P , [C 0 P intersect circle (P ∗ , 2Rp ) at B 00 , C 00 , respectively. Prove that BB 00 , CC 00 intersect on circle (P ∗ , 2Rp ). Problem 19. Given triangle ABC and A1 B1 C1 is cevian triangle of an arbitrary point P . P ∗ is isogonal conjugate of P . A2 , B2 , C2 are reflections of P through B1 C2 , C1 A1 , A1 B1 , respectively. AA2 , BB2 , CC2 are concurrent at BP . Prove that circumcenter of triangle A2 B2 C2 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. A1 B1 C1 is the pedal triangle of P wrt ∆ABC, A2 B2 C2 is the circumcevian triangle of Q wrt ∆ABC. Show that (P A1 A2 ), (P B1 B2 ), (P C1 C2 ) 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 MN is always through a fix point when P , Q move. Problem 22. Let ABC be a triangle inscribed (O). AA0 , BB 0 , CC 0 are altitudes. AA0 intersect (O) at second point D. E is a point on A0 B 0 such that BE ⊥ OA. DB 0 intersect (O) at second point F . BF intersects AE at K. Prove that K is midpoint of B 0 C 0 . 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). Ka is reflection of O through BC. (Ka ) is circle center Ka and passes through B, C. (Oa ) touchs AB, AC and touchs (Ka ) externally at A0 (Oa is inside triangle ABC). Similarly we have B 0 , C 0 . Prove that AA0 , BB 0 , CC 0 are concurrent. Problem 25. Let ABC be a triangle. A circle (ω) pass through B, C intersect AC, AB at E, F . BE intersects CF at I. AI intersect circle (ω) at Z (the point Z is inwardly to triangle ABC). BE, CF intersect circumcircle (ACF ), (ABE) at M, N, resp (M, N are outwardly to triangle ABC). Let the points M 0 ≡ AB ∩ ZM, N 0 ≡ AC ∩ ZN and K ≡ MN 0 ∩ M 0 N. Prove that KM = KN. 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, MN, 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 = CE. Problem 28. Let AB be a segment and P , P 0 are two arbitrary points. I is a point on AB. M, N lie on P A, P B, resp, such that IM k P B, MN k AB. M 0 , N 0 lie on P 0 B, P 0 A, resp, such that IM 0 k P 0 A, M 0 N 0 k AB. K lies on circumcircle (IMN) such that MK k IN 0 . L lies on circumcircle (IM 0 N 0 ) such that M 0 L k IN. Prove that I, K, L are collinear.

Tran Quang Hung - Red geometry

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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 k 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 6≡ 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 , C1 . A2 , B2 , C2 lies on IA1 , IB1 , IC1 such that IA2 . IA1 = IB2 . IB1 = IC2 . IC1 . O is circumcenter of triangle ABC. OA2 , OB2 , OC2 cuts circumcircle (AIA2 ), (BIB2 ), (CIC2 ) again at A3 , B3 , C3 . Prove that AA3 , BB3 , CC3 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 0 , B 0 , Q, resp. Prove that P Q, BB 0 , CC 0 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), (OCD) 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. A0 , B 0 , C 0 are reflections of P through BC, CA, AB, resp. QA0 , QB 0 , QC 0 cuts BC, CA, AB at D, E, F , resp. Prove that AD, BE, CF are concurrent. Problem 35. Let ABC be a triangle and A0 B 0 C 0 is cevian triangle of a point P . Intersections of B 0 C 0 , BC; C 0 A0 , CA; A0 B 0 , AB are collinear on d. DEF is cevian triangle of a point Q with respect to A0 B 0 C 0 . B 0 E, C 0 F cut d at M, N, resp. MB cuts NC at R. AR cuts B 0 C 0 at T . T M, T N cut A0 C 0 , A0 B 0 at K, L, resp. Prove that B 0 K, C 0 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 Pa , Pb , Pc , resp. Let `a , `b , `c , be tangets of circumcircle (P BC), (P CA), (P AB) at Pa , Pb , Pc , 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, a1 , a2 , a3 be three three lines concur at A. p intersect a1 , a2 , a3 at P1 , P2 , P3 , q intersect a1 , a2 , a3 at Q1 , Q2 , Q3 , r intersect a1 , a2 , a3 at R1 , R2 , R3 , resp. P3 Q2 cuts r at K. L is a point such that (P2 Q2 AL) = −1. Assume that Q1 R2 , P2 R3 and OA are concurrent. Prove that R1 P2 , Q2 R3 and KL are concurrent. Problem 38. Let ABCD be cyclic quadrilateral with circumcircle (O). l is a tangent of (O). l1 , l2 , l3 , l4 are reflections of l through AB, BC, CD, DA, resp. Prove that there are four cyclic points in six intersections of l1 , l2 , l3 , l4 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 . P1 , P2 , P3 , P4 are intersections of l with circumcircle (P AB), (P BC), (P CD), (P DA), resp. l1 , l2 , l3 , l4 is tangent of circumcircles (P AB), (P BC), (P CD), (P DA) at P1 , P2 , P3 , P4 , resp. Prove that there are four cyclic points in six intersection of l1 , l2 , l3 , l4 .

Tran Quang Hung - Red geometry

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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 MNKL is cyclic and MK, 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. MT1 , MT2 are tangent of (AEF ) at T1 , T2 , resp. I, K are midpoint of MT1 , MT2 , 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 ME, MB and (O) internally, (O2 ) touches NF , ND 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 O1 R, ∠QO2 S, ∠O1 OO2 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 ME always passes throuh centroid G of triangle ABC. Problem 44. Let ABC 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 0 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 0 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 0 are two isogonal conjugate point with respsect to ABC. K, K 0 are projection of P and P 0 on line BC, resp. AH is altitude of ABC. A1 , A2 ∈ AH such that AA1 = P K, AA2 = P 0 K 0 . P 0A1 , P A2 intersect line BC at P , Q. Prove that BP = CQ. Problem 49. Let ABC be a triangle and XY Z is pedal triangle of a point P with respect to ABC. P 0 is isogonal conjugate of P . (O, R) is circumcircle of triangle XY Z. C is circle (P , 2R). Ray Y P 0, ZP 0 intersects C at M, N, resp. MN cuts Y Z at R. T is projection of R on P A. Prove that T is inversion of A with respect to C.

Tran Quang Hung - Red geometry

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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 XY . 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 = BQ. 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 0 , Y 0 , Z 0 are reflections of X, Y , Z through OP . Prove that AX 0 , BY 0 , CZ 0 and OP are concurrent. Problem 52. Let ABC be a triangle inscribed (O). Pa Pb Pc is pedal triangle of point P such that circumcenter of (Pa Pb Pc ) lies on OP . P A, P B, P C intersect (O) again at X 0 , Y 0 , Z 0 . X, Y , Z are reflections of P through BC, CA, AB. XX 0 , Y Y 0 , ZZ 0 intersect (O) again at A0 , B 0 , C 0 . Let Pa0 Pb0 Pc0 be pedal triangle of P with respect to triangle A0 B 0 C 0 . Prove that the triangles Pa Pb Pc and Pa0 Pb0 Pc0 have the same circumcircle. Problem 53. Let ABC be a triangle and the points M, N, P . BC, CA, AB intersect MN at A0 , B 0 , C 0 . P A intersect MN at A1 . Let A2 be a point such that cross ratio (MNC 0 A1 ) =(NMB 0 A2 ). Defined similarly B2 , C2 . Prove that AA2 , BB2 , CC2 are concurrent. Let ABC be a triangle and a point P . A1 B1 C1 is pedal triangle of P . P ∗ is isogonal conjugate of P . A2 B2 C2 is pedal triangle of P ∗ . Q is a point on line P P ∗. A2 Q, B2 Q, C2 Q cut cirumcircle (A1 B1 C1 ) again at A3 , B3 , C3 , respectively. a) Prove that A1 A3 , B1 B3 , C1 C3 are concurrent on line P P ∗ . b) Prove that AA3 , BB3 , CC3 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. MI 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 . A0 B 0 C 0 is pedal triangle of P with respect to triangle ABC. O is circumcenter of triangles ABC, (O 0) is circumcircle of triangle A0 B 0 C 0 . P A0 , P B 0 , P C 0 intersects (O 0 ) again at A1 , B1 , C1 , respectively. Assume that P , O, O 0 are collinear. Prove that circumcirles (P AA1 ), (P BB1 ), (P CC1) have a common point other than P . Problem 57. Let ABC 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 KO1 O2. P is point on (L). The line passes though P and perpendiculer to OP intersects (O) at B 0 , C 0 . Prove that nine−point center of triangle AB 0 C 0 always lies on a fixed circle (J) and LJ ⊥ EF .

Tran Quang Hung - Red geometry

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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 k 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). P1 , P2 are two points on OI. AP1 cuts (O) again at A1 . A1 P2 cuts (O) again at A2 . Similarly we have B1 , B2 , C1 , C2 , D1 , D2 , E1 , E2 , F1 , F2 . Prove that A2 D2 , B2 E2 , C2 F2 , 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 AEF . Ray IA 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 A0 on (O). b) Circumcircles (BF H), (CEH) cut BA0 , CA0 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 KMN. Prove that IH and tangent at B, C of (O) are concurrent. Let ABC be a triangle and A0 B 0 C 0 is pedal triangle of a point P . AA0 cuts P B 0 , P C 0 at M, N. The lines pass through M, N and parallel to B 0 C 0 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 k AB, LP k 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 HK . a) Prove that HK K and AO intersect on (O). b) OK is isogonal conjugate of HK 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 k BE, OK N k CF . Prove that LN k BC. 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 AHK . e) Q, R lie on BE, CF , resp such that P Q k AB, P R k AC. Prove that QR k BC. f ) Prove that NQ, LR and AHK are concurrent. g) D is projection of K on AHK . Prove that DK, EF , BC are concurrent. h) Prove that KN ⊥ BE, KL ⊥ CF . i) Prove that nine points D, E, F ; P , Q, R; K, L, N lie on a circle (NK ). j) Prove that NK is midpoint of P K and KNK is parallel to AO. k) Prove that HK , NK , O are collinear.

Tran Quang Hung - Red geometry

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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 , C1 . A2 B2 C2 is pedal triangle of P , assume that AA2 , BB2 , CC2 are concurrent. Let A3 ∈ B1 C1 , B3 ∈ C1 A1 , C3 ∈ A1 B1 such that P A3 ⊥ BC, P B3 ⊥ CA, P C3 ⊥ AB. Prove that AA3 , BB3 , CC3 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. MK cuts circumcircle (KEF ) again at N. Prove that ∠LAB = ∠NAC. Problem 66. Let ABC be a triangle and A1 B1 C1 is pedal triangle of a point P with respect to triangle ABC. T is a point on circumcircle (A1 B1 C1 ). ` is line passing though T and perpendicular to P T . A2 , B2 , C2 lie on ` such that P A2 ⊥ P A, P B2 ⊥ P B, P C2 ⊥ P C. Prove that AA2 , BB2 , CC2 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 0, K 0 are circumcenter and symmedian point of triangle AEF , resp. Prove that OK, O 0K 0 and BC are concurrent. Problem 69. Let ABC be a triangle with circumcircle (O). D, E lie on (O). Circle (C1 ) pass through A, D and tangent to AC. Circle (C2 ) pass through A, E and tangent to AB. (C1 ) cuts (C2 ) 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. MN 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 perpendicular 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) UV cuts KL at S. (O2 ) is circumcircle of triangle SKU, (O3) is circumcircle of triangle SLV . 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 Q at 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 . MN 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 MN passing through H. K is in BH such that MK is tangent of circumcircle (F MH). L is in CH such that NL is tangent of circumcircle (ENH). Prove that KL k BC.

Tran Quang Hung - Red geometry

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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 A0 . B 0 , C 0 . Tangent at A of (O) cuts BC at T . T P cuts (O) at M, N. Prove that triangle A0 B 0 C 0 and A0 MN have the same A0 −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 MN k AQ. Problem 76. Let ABC 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) 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 77. Let ABC be triangle and a point P . A1 B1 C1 is pedal triangle of P with respect to triangle ABC. A2 B2 C2 is circumcevian triangle of P . A3 B3 C3 is pedal triangle of P with respective to triangle A2 B2 C2 . Prove that A1 B1 C1 and A3 B3 C3 are persective if only if A1 B1 C1 and A2 B2 C2 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 0 . Prove that intersecton of tangents at M, M 0 of (ω) lies on symmedian of triangle ABC. c) Let N, N 0 is isogonal conjugate of M, M 0 with respect to triangle ABC. Prove that tangents at N, N 0 of circumcircle (ANN 0 ) intersects on line BC. d) Prove that midpoint of NN 0 lies on nine−points circle of triangle ABC. Problem 79. Let ABCD be circumscribed quadrilateral with incenter I. O1, H1 is circumcenter −−−→ −−−→ and orthocenter of triangle IAB. K1 is a point such that O1 K1 = k O1 H1 , k is a fixed constant, d1 is the line passing through K1 and perpendicular to AB. Similarly we get d2 , d3 , d4 . Prove that d1 , d2 , d3 , d4 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 MN k P A. Q is reflection of P through MB, R is reflection of P through NC. Prove that QR ⊥ BC. 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 ∠BCD. 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 0 . DQ cuts (O) again at E. EP ∗ cuts (O) again at E 0 . Prove that AE k D 0 E 0 .

Tran Quang Hung - Red geometry

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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 ⊥ BC. EP cuts BC at F and (O) second time at G. Q is a point on \ tan F QB tan P[ AB = . AG such that F Q k AP . Prove that tan P[ AC 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 MN, 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 k DE, H lies on BD such that CH k AF . AF ∩ DE ≡ X, DE ∩ CH ≡ Y , CH ∩ GB ≡ Z, GB ∩ AF ≡ T . M lies on AC. N lies on BD such that MN k AB, P lies on AC such that NP k BC, Q lies on BD such that P Q k CD. dM passes though M, dP passes though P such that dM k dP k DE. dQ passes though Q, dN passes though N such that dQ k dN k AF . dQ ∩ dM ≡ U, dM ∩ dN ≡ V , dN ∩ dP ≡ W , dP ∩ dQ ≡ 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 k EF . 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 to triangle 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. A1 B1 C1 is pedal triangle of P with respect to triangle ABC. A2 , B2 , C2 are symmetric of Q through A1 , B1 , C1 , respectively. A3 , B3 , C3 are reflection of A2 , B2 , C2 through BC, CA, AB , respectively. Prove that Q, A3 , B3 , C3 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 , C1 . A2 , B2 , C2 is pedal triangle of P with respect to triangle ABC. H is orthocenter of triangle ABC. A3 , B3 , C3 are symmetric of H thourgh A2 , B2 , C2 respectively. Prove that A1 A3 , B1 B3 , C1 C3 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 UI, JV , respectively. Prove that P Q, KL, NP are concurrent.

Tran Quang Hung - Red geometry

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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 (K1 ), (K2 ), (K3 ), (K4 ), (K5 ) as in figure. Prove that XM, Y N, ZP , T Q, UR 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. NB cuts P F at K, NC cuts P E at L, CK cuts BL at Q. Prove that P Q k 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 DCE, 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 IAB, 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 (K1 ), (K2 ), (K3 ), (K4 ), (K5 ), (K6 ) as in figure. Let G, H, J, L, M, N be incenters of triangles KAB, KBC, KCD, KDE, KEF , 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 MNP . Let Fe, G be Feuerbach points of triangle ABC and MNP . Prove that OI passes through midpoint of FeG. 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 PE PR = . at Q, R. Prove that PQ PF 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 O1 O2 , respectively. IA, IB cut O1 O2 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. MO1 cuts KO2 at T . Prove that T M = T K.

Tran Quang Hung - Red geometry

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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 ⊥ ST . 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 , C1 are midponts of BC, CA, AB, resp. P A, P B, P C cut (O) again at A2 , B2 , C2 , resp. A2 A1 , B2 B1 , C2 C2 cut (O) again at A3 , B3 , C3 . A3 Q, B3 Q, C3 Q cut BC, CA, AB at A4 , B4 , C4 . Prove that AA4 , BB4 , CC4 are concurrent. Problem 105. Let ABCD be circumscribed quadrilateral and M is its Miquel point. (M) is a circle center M. Let A0 , B 0 , C 0 , D 0 invert A, B, C, D through (M), respectively. Prove that A0 B 0 C 0 D 0 is circumscribed quadrilateral. Problem 106. Let ABC and A0 B 0 C 0 be two triangles inscribed circle (O). Prove that orthopoles of B 0 C 0 , C 0 A0 , A0 B 0 with respect to triangle ABC and orthopoles of BC, CA, AB with respect to triangle A0 B 0 C 0 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 to triangle ABC. M, N are on line EF such that BM k CN. Let Q be a point such that P BQC is a prallelogram. BQ cuts AC at K, CQ cuts AB at L. a) Prove that MK, NL and BC are concurrent at a point T . b) Prove that if BM k CN k 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 k AB, ON k 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 A0 . Similarly we have B 0 , C 0 . Prove that AA0 , BB 0 , CC 0 are concurrent. Problem 113. Let ABC 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 MN 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.

Tran Quang Hung - Red geometry

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Problem 114. P , Q are antigonal conjugates with respect to 4ABC. Then 9−point circles of 4AP Q, 4BP Q, 4CP 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 A0 , B 0 , C 0 , respectively. Prove that AA0 , BB 0 , CC 0 are concurrent. Problem 116. Let ABC be a triangle, orthocenter H and a point P . Let A0 B 0 C 0 be pedal triangle of P . (E) is circumellipse of triangle A0 B 0 C 0 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 ABC 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 A0 , B 0 , C 0 , respectively. Prove that A0 X, B 0 Y , C 0 Z are concurrent.