Problem

Source: VMO 2021 P7 Vietnam National Olympiad

Tags: geometry, Concyclic, collinear, circumcircle



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