Triple Massey products on curves, Fay's trisecant identity and tangents to the canonical embedding

We show that Fay's trisecant identity follows from the $A_\infty$-constraint satisfied by certain triple Massey products in the derived category of coherent sheaves on a curve. We also deduce the matrix analogue of this identity that can be conveniently formulated using quasideterminants of matrices with noncommuting entries. On the other hand, looking at more special Massey products, we derive a formula for the tangent line to a canonically embedded curve at a given point.