Question 14
Let $ABC$ be an acute, non isosceles triangle with circumcircle $(O)$ and circumradius $R$. Denote $M$, $N$, $P$ as midpoints of segments $BC$, $CA$, $AB$ respectively. Let $(\omega)$ be the circle passes through $A$, $O$ and tangent to $OM$. Circle $(\omega)$ meets $AB$, $AC$ at $E$, $F$. Denote $I$ as midpoint of segment $EF$ and $K$ as the intersection of $EF$, $NP$. Prove that $R = 2IK$ and $IMO$ is an isosceles triangle.