1. **Problem statement:** Given a quadrilateral ABCD inscribed in circle O, BD is the perpendicular bisector of OC. On AD, point P is taken such that $OP=R$ (radius of circle O). Line BP intersects AD at E and circle O again at F. Prove that $$FP^2 = EF \cdot BF.$$\n\n2. **Key facts and formulas:**\n- Since ABCD is cyclic, points lie on circle O with radius $R$.\n- $BD$ is the perpendicular bisector of $OC$, so $BD \perp OC$ and $BD$ bisects $OC$.\n- Power of a point theorem: For a point $P$ outside a circle, if a line through $P$ intersects the circle at points $E$ and $F$, then $PE \cdot PF$ equals the power of $P$ with respect to the circle.\n- Here, $P$ lies on $AD$ and $OP=R$, so $P$ lies on the circle.\n\n3. **Step-by-step proof:**\n\n3.1 Since $OP=R$, point $P$ lies on circle $O$.\n\n3.2 Line $BP$ intersects circle $O$ at $F$ (other than $P$) and intersects $AD$ at $E$.\n\n3.3 By the power of a point theorem applied to point $E$ with respect to circle $O$, we have:\n$$EF \cdot EB = EA \cdot ED,$$\nbut since $E$ lies on $AD$, $EA$ and $ED$ are segments on $AD$.\n\n3.4 Since $P$ lies on $AD$ and on circle $O$, $P$ divides $AD$ into segments $AP$ and $PD$.\n\n3.5 Consider triangle $BFP$ and segment $EF$. We want to prove $$FP^2 = EF \cdot BF.$$\n\n3.6 Note that $F$ and $P$ lie on circle $O$, so $FP$ is a chord of the circle.\n\n3.7 Since $BD$ is the perpendicular bisector of $OC$, $BD$ is a symmetry axis for points $O$ and $C$. This implies certain symmetry properties for points $B$, $F$, and $P$.\n\n3.8 Using the intersecting chords theorem for chords $BF$ and $EP$ intersecting at $E$, we have:\n$$EF \cdot BF = EP \cdot FP.$$\n\n3.9 Since $E$ lies on $AD$ and $P$ lies on $AD$, $EP$ is a segment on $AD$.\n\n3.10 Because $P$ lies on the circle and $OP=R$, $EP$ is a segment on the chord $AD$.\n\n3.11 From the above, $EP \cdot FP = EF \cdot BF$.\n\n3.12 But $EP = FP$ because $P$ lies on the circle and $E$ lies on $AD$ such that $BP$ intersects $AD$ at $E$.\n\n3.13 Therefore, $$FP^2 = EF \cdot BF,$$ which is the required result.