CWMO 2012 Problem 4

$P$ is a point in the $\Delta ABC$ , $\omega$ is the circumcircle of $\Delta ABC$ . $BP \cap \omega = \left\{ {B,{B_1}} \right\}$ , $CP \cap \omega = \left\{ {C,{C_1}} \right\}$ , $PE \bot AC$ ， $PF \bot AB$ . The radius of the inscribed circle and circumcircle of $\Delta ABC$ is $r,R$ . Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$ .

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