1977 AMC 12 Problema 19

Two adjacent circumcenters lie on the perpendicular bisector of the side shared by their triangles

Use the fact that $A,$ $E,$ $C$ are collinear and $B,$ $E,$ $D$ are collinear

Both $P$ and $Q$ lie on the perpendicular bisector of $BE,$ while both $R$ and $S$ lie on the perpendicular bisector of $DE.$ Since $B,$ $E,$ $D$ are collinear, these bisectors are parallel, so $PQ\parallel RS.$ Likewise, $Q,$ $R$ lie on the perpendicular bisector of $CE,$ and $S,$ $P$ lie on that of $AE.$ Since $A,$ $E,$ $C$ are collinear, $QR\parallel SP.$ Therefore $PQRS$ is always a parallelogram.

Therefore, the correct answer is A.