Question 5
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 midpoints of $BE$ and $CF$ cuts $BF$, $CE$ at $M, N$ respectively. Prove that the points $A, D, M$ and $N$ lie on a same circle.
b) Lines $AD, AG$ intersect the circumcircle of $DBC, GBC$ 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.
a) The line passing through the midpoints of $BE$ and $CF$ cuts $BF$, $CE$ at $M, N$ respectively. Prove that the points $A, D, M$ and $N$ lie on a same circle.
b) Lines $AD, AG$ intersect the circumcircle of $DBC, GBC$ 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.