Question 149
Let $ABC$ be an acute, non-isosceles triangle inscribed in a circle $(O)$ and $M$ is the midpoint of $BC$. Take $D$ on the segment $BC$ and the circle $(ABD)$ meets $AC$ again at $E$, the circle $(ACD)$ meets $AB$ again at $F$. Lines $BE$, $CF$ meet at $K$ and lines $AK$, $EF$ meet at $R$. Suppose that there exist the circle passing through $B$, $C$ and external tangent to $(AEF)$ at $T$. Prove that $RT$ bisects $BC$.