Let $\Delta ABC$ be an acute triangle with $a>b$, center $O$ of its circumscribed circle and middle point $M$ of $AC$. Let $K$ be the reflection of $O$ in $M$. Point $E\in BC$ is such that $EO\perp AB$. Point $F\in MK$ is such that $FK=OE$ and $K$ lies between $F$ and $M$. The altitude through $C$ and the angle bisector of $\angle CAB$ intersect in $D$. Let $BD$ intersect the circumscribed circle of $\Delta ABC$ for a second time in $P$. Prove that $AP\perp CF$.