  We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the
discrete Pompeiu property with respect to isometries (similarities), if,
whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of
$f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically
zero. We show that every parallelogram and every quadrangle with rational
coordinates has the discrete Pompeiu property w.r.t. isometries. We also
present a family of quadrangles depending on a continuous parameter having the
same property. We investigate the weighted version of the discrete Pompeiu
property as well, and show that every finite linear set with commensurable
distances has the weighted discrete Pompeiu property w.r.t. isometries, and
every finite set has the weighted discrete Pompeiu property w.r.t.
similarities.
