Question 5
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies $$
\angle PBC=\angle DBA \quad\text{and}\quad \angle PDC=\angle BDA.
$$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
\angle PBC=\angle DBA \quad\text{and}\quad \angle PDC=\angle BDA.
$$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.