Question 4
Triangle $ABC$ is inscribed in a fixed circle $(O)$, with $B,C$ fixed and $A$ moving on $(O)$ so that $AB>BC$. Let $M$ be the midpoint of $AC$. The circle with diameter $BM$ meets $(O)$ again at $R$. The line $RM$ meets $(O)$ again at $Q$ and meets $BC$ at $P$. The circle with diameter $BP$ meets $AB$ and $BO$ again at $K$ and $S$, respectively.
(a) Prove that $SR$ passes through the midpoint of $KP$.
(b) Let $N$ be the midpoint of $BC$. The radical axis of the circles with diameters $AN$ and $BM$ meets $SR$ at $E$. Prove that $ME$ always passes through a fixed point.
(a) Prove that $SR$ passes through the midpoint of $KP$.
(b) Let $N$ be the midpoint of $BC$. The radical axis of the circles with diameters $AN$ and $BM$ meets $SR$ at $E$. Prove that $ME$ always passes through a fixed point.