LDP polygons and the number 12 revisited

We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $area(\mathrm{\Delta })+area({\mathrm{\Delta }}^{\ast })=6$ for reflexive $\mathrm{\Delta }$. The identity is equivalent to the stringy Libgober-Wood identity for toric log del Pezzo surfaces.