Question 1
Let $ABCD$ be a cyclic quadrilateral. The extensions of $AB$ and $DC$ meet at $E$ and the extensions of $AD$ and $BC$ meet at $F$. The bisectors of $\angle BEC$ and $\angle DFC$ meet at $P$ which lies on $AC$. Let $PF$ intersect $CD$ at $U$ and $PE$ intersect $BC$ at $V$. Prove that:
(a) $DB$ is parallel to $UV$.
(b) The lines $AC$ and $EF$ intersect at the midpoint of $EF$.
(a) $DB$ is parallel to $UV$.
(b) The lines $AC$ and $EF$ intersect at the midpoint of $EF$.