  The point-line geometry known as a \textit{partial quadrangle} (introduced by
Cameron in 1975) has the property that for every point/line non-incident pair
$(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in
particular, the well-studied objects known as \textit{generalised quadrangles}
are each partial quadrangles. An \textit{intriguing set} of a generalised
quadrangle is a set of points which induces an equitable partition of size two
of the underlying strongly regular graph. We extend the theory of intriguing
sets of generalised quadrangles by Bamberg, Law and Penttila to partial
quadrangles, which surprisingly gives insight into the structure of hemisystems
and other intriguing sets of generalised quadrangles.
