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Archive for Tháng Mười Hai, 2009

5 concurrent lines

Given a cyclic quadrilateral ABCD and its circumcircle (O). Denote P the intersection of AC and BD, I_1, I_2, I_3, I_4 the incenters of triangles ABP, BCP, CDP, DAP, respectively. Let \omega_1 be the circle which is tangent to rays PA,PB and (O) at X_1. Similarly we define X_2,X_3,X_4. Prove that five lines I_1X_3, I_2X_4, I_3X_1,I_4X_2 and OP are concurrent.

Proof: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=315105&p=1923707&hilit=russian+olympiad#p1923707

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Some problems of me about coaxal circles.

Problem 1: Given triangle ABC and its incircle (I). (I) touches BC, CA, AB at A_1,B_1,C_1, respectively. Let A_2,B_2,C_2 be the points on IA_1,IB_1,IC_1 such that \frac {\overline{IA_2}}{\overline{IA_1}} = \frac {\overline{IB_2}}{\overline{IB_1}} = \frac {\overline{IC_2}}{\overline{IC_1}}. Prove that 3 circles (AIA_2),(BIB_2), (CIC_2) concur at the point Q\neq I.


Problem 2: Given triangle ABC and its circumcenter O. Denote P a point such that the pedal triangle A'B'C' of P wrt \Delta ABC is the cevian triangle of point Q wrt \Delta ABC. Prove that (APA'),(BPB') and (CPC') concur at a point which lies on OP.


Problem 3: Given triangle ABC and its incircle (I). Let A_1,B_1,C_1 be the intersections of AI and BC,BI and CA,CI and AB, respectively; A_2 be the intersection of a line through I and perpendicular to AI and BC, similarly define B_2, C_2. Show that (AA_1A_2), (BB_1B_2), (CC_1C_2) concur at two points.


Problem 4: Given triangle ABC and its circumcircle (O). Denote M_a, M_b, M_c the midpoints of BC, CA, AB, respectively, P an arbitrary point inside \Delta ABC, A',B',C' the second intersections of AP, BP, CP and (O). Prove that (A'OM_a), (B'OM_b), (C'OM_c) concur at two points.


Problem 5: Given triangle ABC, its incircle (I) and its circumcircle (O). Denote A_1 be the intersection of the line through  I and perpendicular to AI and BC, similarly define B_1,C_1.  Denote A_2,B_2,C_2 the projections of I on BC,CA,AB, respectively. Prove that three circles (AA_1A_2), (BB_1B_2), (CC_1C_2) are coaxal.

Problem 6: Given a triangle ABC with its circumcenter O. Let d be an arbitrary line in the plane. d intersects BC, CA, AB at X, Y, Z, respectively. Let H be the projection of O on d. Prove that 3 circles (AXH), (BYH), (CZH) are coaxal.


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Near Hagge circle 2

Given triangle ABC , its orthocenter H and its circumcircle (O). Denote A_1,B_1,C_1 the projections of A,B,C on BC,CA,AB, respectively; P an arbitrary point on OH. AP,BP,CP intersect (O) again at A_2,B_2,C_2, respectively; A_3,B_3,C_3 the reflections of A_2,B_2,C_2 wrt A_1,B_1,C_1, respectively. Prove that 4 points H,A_3,B_3,C_3 are concyclic and its circumcenter lies on OH.

The solution of the generalization can be found at here


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Near Hagge circle 1

Given triangle ABC and its orthocenter H, its circumcircle (O). Let P be an arbitrary point on the plane. AP,BP,CP intersects (O) again at A_1,B_1,C_1, respectively. Let A_2B_2C_2 be the pedal triangle of P wrt \Delta ABC, A_3,B_3,C_3 be the reflections of A_1,B_1,C_1 wrt A_2,B_2,C_2 ,respectively. Prove that 4 points H,A_3,B_3,C_3 are concyclic.


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