 Elliptic collinearity determinant

The determinant of the columns $(1,\wp (z),{\wp }^{\prime }(z))$, $(1,\wp (w),{\wp }^{\prime }(w))$ and $(1,\wp (-z-w),{\wp }^{\prime }(-z-w))$ vanishes identically. Its apparent cubic poles at $z=0,-w$ have order at most two because the coefficient multiplying the cubic-pole entry vanishes linearly. Column coincidences give six zeros, unless $3w$ is a lattice point, when they give four zeros with one multiple zero. In the half-period case, the two apparent zero locations at poles are removable zeros: ${\wp }^{\prime }(w)=0$ and the two $1/z$ terms cancel, leaving $O(z)$. Thus the zero count exceeds possible pole order in every case. The argument principle proves the identity, which expresses collinearity of the three points on the Weierstrass equation of an elliptic curve.