Question 83
$ABC$ is a triangle with $BC \gt CA \gt AB$. $D$ is a point on side $BC$, and $E$ is a point on $BA$ produced beyond $A$ so that $BD = BE = CA$. Let $P$ be a point on side $AC$ such that $E, B, D, P$ are concyclic, and let $Q$ be the second intersection point of $BP$ with the circumcircle of $\triangle ABC$. Prove that $AQ + CQ = BP$.