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concyclic and collinear wanted, 5 circles related, starting with circumcircle

Source: VMO 2021 P7 Vietnam National Olympiad

December 26, 2020
geometryConcycliccollinearcircumcircle

Problem Statement

Let ABC ABC be an inscribed triangle in circle (O) (O) . Let D D be the intersection of the two tangent lines of (O) (O) at B B and C C . The circle passing through A A and tangent to BC BC at B B intersects the median passing A A of the triangle ABC ABC at G G . Lines BG,CG BG, CG intersect CD,BD CD, BD at E,F E, F respectively. a) The line passing through the midpoint of BE BE and CF CF cuts BF,CE BF, CE at M,N M, N respectively. Prove that the points A,D,M,N A, D, M, N belong to the same circle. b) Let AD,AG AD, AG intersect the circumcircle of the triangles DBC,GBC DBC, GBC at H,K H, K respectively. The perpendicular bisectors of HK,HE HK, HE, and HFHF cut BC,CA BC, CA, and ABAB at R,P R, P, and QQ respectively. Prove that the points R,P R, P, and QQ are collinear.
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