VMO 2021 Problem 7

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.