Question 49
Let a line $l$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that
$$
FB \cdot FA + CP^2 = CF^2 \iff AD \cdot BE = PD^2.
$$
$$
FB \cdot FA + CP^2 = CF^2 \iff AD \cdot BE = PD^2.
$$