Let $O_BF$ $\cap$ $O_CD = T$. It suffices to prove $O_A, T, E$ are collinear. Claim 1: $FDO_BO_C$ is a parallelogram. Proof$CF \parallel DE , \angle CFO_C = \angle EDO_B \Longrightarrow FO_C || DO_B$. Since the perpendiculars from $O_C$ and $O_D$ are equal and parallel, we get $O_CO_B \parallel BC \Longrightarrow O_CO_B \parallel FD$. $Q.E.D. \clubsuit$ Claim 2: $\square O_ADTF \cong \square EO_CTO_B$ Proof$O_AD=O_CE , O_BE = O_AF , FD = O_CO_B \Longrightarrow \triangle FO_AD \cong \triangle O_BEO_C$. And we also have $\triangle FTD \cong \triangle O_BTO_C \Longrightarrow \square O_ADTF \cong \square EO_CTO_B$. $Q.E.D. \clubsuit$ Since $\square O_ADTF \cong \square EO_CTO_B$, we have $\angle O_ATD = \angle ETO_C \Longrightarrow O_A, E, T$ are collinear. $Q.E.D. \clubsuit$