  We consider geometric problems on planar $n^2$-point sets in the congested
clique model. Initially, each node in the $n$-clique network holds a batch of
$n$ distinct points in the Euclidean plane given by $O(\log n)$-bit
coordinates. In each round, each node can send a distinct $O(\log n)$-bit
message to each other node in the clique and perform unlimited local
computations. We show that the convex hull of the input $n^2$-point set can be
constructed in $O(\min\{ h,\log n\})$ rounds, where $h$ is the size of the
hull, on the congested clique. We also show that a triangulation of the input
$n^2$-point set can be constructed in $O(\log^2n)$ rounds on the congested
clique. Finally, we demonstrate that the Voronoi diagram of $n^2$ points with
$O(\log n)$-bit coordinates drawn uniformly at random from a unit square can be
computed within the square with high probability in $O(1)$ rounds on the
congested clique.
