[desertrider12]

Any time the line is tangent to the curve, you treat that one point as a double intersection. You can show this by setting the line equation equal to the elliptic curve and solving for the x coordinates of the 3 intersections. The double intersection's x coordinate is a double root, just like x=0 in x^2 = 0.

A vertical line is perfectly valid (could be tangent or not), and in this case the "sum" is the "infinity point", which isn't any specific location in the grid but is a useful idea to make the math work. The infinity point is the additive identity: P + inf = P. For the amount of detail in this article, it should have mentioned this.