  Let $S\subset \mP^4$ be a general K3 surface of degree 6 and genus 4. In this
paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to
$S$ whose general point is a plane that intersects $S$ in a curvilinear scheme
of length six supported at three non collinear points. The variety $X_S$ can be
identified as the relevant part of the fixed locus of the so called
\emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of
0--dimensional schemes of length three of $S$. In this paper we prove that: (a)
$X_S$ has dimension 3, is irreducible and smooth, except for 210 points that
are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in
the Grassmannian $\mathbb G(2,4)\subset \mP^9$ of planes in $\mP^4$, has degree
$152$.
