Dear Darij
> Please help me with the following problem (I need it as
> a lemma):
>
> Given a cyclic quadrilateral ABCD with the circumcenter
> O. The perpendicular to BD through B meets the
> perpendicular to AC through C at E. The perpendicular
> to BD through D meets the perpendicular to AC through A
> at F. Finally, let X be the intersection of the lines
> AB and CD. Then, the points O, E, F, X are collinear.
>
> Well, it is easy to show that O is the midpoint of EF;
> hence, O, E, F are collinear, but how about X ?

Let A' and D' be the reflections of A and D about O.
O = AA' inter DD', E = CA' inter BD' and X = AB inter DC are
collinear by Pascal's theorem.
Friendly. Jean-Pierre