The associativity of the addition on an elliptic curve is usually proved using Cayley–Bacharach theorem, which is fine as long as there are nine distinct points to work with.
The case when some of these points are equal is rarely taken into account and doesn't seem straightforward to me. Is there a not-too-advanced way to deal with it?
Terence Tao here mentions ‘a perturbation (or Zariski closure) argument’ to fix up the proof. What does that mean exactly?