  We prove discrete Helly-type theorems for pseudohalfplanes, which extend
recent results of Jensen, Joshi and Ray about halfplanes. Among others we show
that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if
every triple of pseudohalfplanes has a common point in $P$ then there exists a
set of at most two points that hits every pseudohalfplane of $\cal H$. We also
prove that if every triple of points of $P$ is contained in a pseudohalfplane
of $\cal H$ then there are two pseudohalfplanes of $\cal H$ that cover all
points of $P$.
  To prove our results we regard pseudohalfplane hypergraphs, define their
extremal vertices and show that these behave in many ways as points on the
boundary of the convex hull of a set of points. Our methods are purely
combinatorial.
