Considering the "visual" method of explaining elliptic curve addition, I wonder what you are supposed to do if one of the points being added is tangent to the curve?
With the classic shape e.g. $y^2 = x^3 - x + 2$ , you could get a horizontal line, that only touches the curve twice, not three times:
The third point can't be $0$, because that would mean that $B=-A$, clearly not the case. It can't be $B$ either, because that would mean that $A=0$ ?