  Gordan and Noether proved in their fundamental theorem that an hypersurface
$X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ has
vanishing hessian (i.e. the determinant of the Hessian matrix). They also
showed that the statement is false if $n\geq 4$, by giving some
counterexamples. Since their proof, several others have been proposed in the
literature. In this paper we give a new one by using a different perspective
which involves the study of standard Artinian Gorenstein $\mathbb{K}$-algebras
and the Lefschetz properties. As a further application of our setting, we prove
that a standard Artinian Gorenstein algebra $R=\mathbb{K}[x_0,\dots,x_4]/J$
with $J$ generated by a regular sequence of quadrics has the strong Lefschetz
property. In particular, this holds for Jacobian rings associated to smooth
cubic threefolds.
