18.21 Let $E$ be the intersection of the diagonals of convex quadrilateral $A B C D$, and let $P, Q, R$ and $S$ be the centers of the circumcircles of $\triangle A B E, \triangle B C E, \triangle C D E$ and $\triangle A D E$, respectively. Then
(A) $P Q R S$ is a parallelogram.
(B) $P Q R S$ is a parallelogram if and only if $A B C D$ is a rhombus.
(C) $P Q R S$ is a parallelogram if and only if $A B C D$ is a rectangle.
(D) $P Q R S$ is a parallelogram if and only if $A B C D$ is a parallelogram.
(E) None of the above is correct.
(28th American High School Mathematics Examination, 1977)
[Solution] The center of the circumcircle of a triangle is the intersection of the perpendicular bisectors of its three sides, so $P, Q, R$ and $S$ are the intersections of the perpendicular bisectors of segments $A E, B E, C E$ and $D E$.

Since lines perpendicular to the same line are parallel (and because lines $A E$ and $C E$ coincide, $B E$ and $D E$ coincide), quadrilateral $P Q R S$ is a parallelogram.
Therefore, the answer is $(A)$.