Question 7
Let $ABC$ be an acute, non-isosceles triangle inscribed in $(O)$. The tangents to $(O)$ at $B,C$ meet at $D$. The circle through $A$ tangent to $BC$ at $B$ meets the median from $A$ again at $G$. The lines $BG,CG$ meet $CD,BD$ at $E,F$, respectively.
a) The line through the midpoints of $BE,CF$ meets $BF,CE$ at $N,M$, respectively. Prove that $A,D,M,N$ are concyclic.
b) The lines $AD,AG$ meet the circumcircles of $DBC,GBC$ again at $H,K$, respectively. The perpendicular bisectors of $HK,HE,HF$ meet $BC,CA,AB$ at $R,P,Q$, respectively. Prove that $R,P,Q$ are collinear.
a) The line through the midpoints of $BE,CF$ meets $BF,CE$ at $N,M$, respectively. Prove that $A,D,M,N$ are concyclic.
b) The lines $AD,AG$ meet the circumcircles of $DBC,GBC$ again at $H,K$, respectively. The perpendicular bisectors of $HK,HE,HF$ meet $BC,CA,AB$ at $R,P,Q$, respectively. Prove that $R,P,Q$ are collinear.