  Tensor diagrams are a handy way to depict complicated relationships between
objects in projective geometry. One of the simpler ones takes two copies of a
$3\times 3$ matrix and computes its adjugate. In this paper, we give a
geometric interpretation of this construction when two different matrices are
used. To do so, we interpret them as coefficient matrices of conic sections in
the projective plane and relate the diagram construction to their common
tangents.
