USAJMO 2011 Question 5

Points $A,B,C,D,E$ lie on a circle $\omega$ and $P$ lies outside it. Lines $PB,PD$ are tangent to $\omega$; $P,A,C$ are collinear; and $DE\parallel AC$. Prove that $BE$ bisects the segment $AC$.

Proposed by Zuming Feng (credit in the supplied notes).