Equidistant circumcenters [self-made]

[Phlembac Adib Hasan]

$ABCD$ is a parallelogram. A line parallel to $AB$ meets line $AD$ at $E$, line $BC$ at $F$. $O_{ABE},O_{FBE},O_{DCE},O_{FCE}$ denote the circumcenter of $\triangle ABE,\triangle FBE,\triangle DCE,\triangle FCE$, respectively. Prove that \[O_{ABE}O_{DCE}=O_{FBE}O_{FCE}.\]
For an additional challenge, prove that the length of $O_{FBE}O_{FCE}$ does not depend on the choice of $EF$.