Question 165
Let $B$, $C$ be two fixed points on the fixed circle $(O)$ ($BC$ is not the diameter of $(O)$). Point $A$ moves on $(O)$ such that $AB \gt BC$ and $M$ is the midpoint of $AC$. The circle of diameter $BM$ intersects $(O)$ at $R$. Suppose that $RM$ intersects $(O)$ at the second point $Q$ and cuts $BC$ at $P$. The circle of diameter $BP$ intersects $AB$, $BO$ at the second points $K$, $S$ respectively.
1. Prove that $SR$ passes through the midpoint of $KP$.
2. Denote $N$ as the midpoint of $BC$. The radical axis of two circles of diameter $AN$, $BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a certain fixed point when $A$ moves on $(O)$.
1. Prove that $SR$ passes through the midpoint of $KP$.
2. Denote $N$ as the midpoint of $BC$. The radical axis of two circles of diameter $AN$, $BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a certain fixed point when $A$ moves on $(O)$.